[Paper Review] Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds

Mehrtash Harandi, Richard Hartley+3 more|arXiv (Cornell University)|Jan 31, 2014

Face and Expression Recognition69 references4 citations

TL;DR

This paper proposes extrinsic methods for sparse coding and dictionary learning on Grassmann manifolds by embedding subspaces into symmetric matrices via an isometric mapping, enabling efficient computation. The approach achieves state-of-the-art classification accuracy on video and image set tasks by leveraging chordal means and kernelized Hilbert space embeddings.

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 over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. 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 closed-form solutions 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 Hilbert spaces. Experiments 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 graph-embedding Grassmann discriminant analysis.

Motivation & Objective

• To address sparse coding and dictionary learning on Grassmann manifolds, where subspaces represent data such as image sets or video sequences.
• To overcome the computational burden of intrinsic Riemannian methods by using extrinsic embedding into symmetric matrices.
• To develop an efficient, atom-by-atom dictionary learning algorithm on Grassmann manifolds.
• To extend the framework to handle non-linear data via kernel-induced Hilbert space embeddings.
• To improve discrimination accuracy in visual recognition tasks compared to existing state-of-the-art methods.

Proposed method

• Embed Grassmann manifolds into the space of symmetric matrices using an isometric mapping that preserves the Frobenius norm, enabling Euclidean-style operations.
• Formulate sparse coding on Grassmann manifolds by minimizing the Frobenius norm between a data subspace and a linear combination of dictionary atoms in the embedded space.
• Use the chordal mean—a closed-form solution based on the projection of the sum of embedded matrices—as the mean of multiple subspaces, ensuring computational efficiency.
• Learn a Grassmann dictionary atom-by-atom by iteratively optimizing the sparse coding objective using the embedded representation.
• Extend the framework to non-linear data by mapping the embedded matrices into a higher-dimensional Hilbert space via a kernel function.
• Leverage the fact that the chordal mean and sparse coding objectives remain computationally tractable in the embedded space due to the isometric property.

Experimental results

Research questions

• RQ1 Can sparse coding on Grassmann manifolds be efficiently performed using extrinsic embedding into symmetric matrices?
• RQ2 How can a Grassmann dictionary be learned in a computationally efficient, atom-by-atom manner?
• RQ3 Does embedding Grassmann data into a Hilbert space via a kernel function improve performance on non-linear data?
• RQ4 How does the proposed method compare to intrinsic Riemannian sparse coding in terms of accuracy and computational cost?
• RQ5 Can the proposed framework achieve state-of-the-art performance on visual recognition tasks involving image sets and video sequences?

Key findings

• The proposed extrinsic method achieves significant improvements in classification accuracy on gender recognition, gesture classification, scene analysis, face recognition, action recognition, and dynamic texture classification.
• The chordal mean provides a closed-form, computationally efficient alternative to iterative Riemannian means on Grassmann manifolds.
• The atom-by-atom dictionary learning algorithm converges faster and scales better than intrinsic methods due to the absence of logarithmic maps.
• Kernelized embedding into Hilbert space enhances performance on non-linear data, outperforming methods like kernelized Affine Hull Method and graph-embedding Grassmann discriminant analysis.
• The isometric embedding preserves the Riemannian structure, enabling exact sparse coding with minimal approximation error.
• Empirical results show consistent gains over state-of-the-art methods across six benchmark visual recognition tasks.