# Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution

arXiv: https://arxiv.org/abs/1310.4891 · HTML: https://mlsanity.com/paper/1310.4891/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2013.

## Built on by (most cited first, 13 of 13)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Multi-manifold Attention for Vision Transformers](https://mlsanity.com/paper/2207.08569/) (2022)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Grassmannian learning mutual subspace method for image set recognition](https://mlsanity.com/paper/2111.04352/) (2021)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [Generalized Relevance Learning Grassmann Quantization](https://mlsanity.com/paper/2403.09183/) (2024)
- [Reconciliation of Statistical and Spatial Sparsity For Robust Image and Image-Set Classification](https://mlsanity.com/paper/2106.00256/) (2021)
- [FICNN: A Framework for the Interpretation of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2305.10121/) (2023)
- [Rapid Grassmannian Averaging with Chebyshev Polynomials](https://mlsanity.com/paper/2410.08956/) (2024)
- [Color Image Set Recognition Based on Quaternionic Grassmannians](https://mlsanity.com/paper/2505.23629/) (2025)
- [Learning Topology-Driven Multi-Subspace Fusion for Grassmannian Deep Network](https://mlsanity.com/paper/2511.08628/) (2025)
- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)
- [Soft-Argmax for the Projective Plane via the Veronese Embedding](https://mlsanity.com/paper/2609.00521/) (2026)

## Abstract (arXiv)

Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse 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, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis.