# Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices

arXiv: https://arxiv.org/abs/1507.02772 · HTML: https://mlsanity.com/paper/1507.02772/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2015.

## Built on by (most cited first, 24 of 24)

- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning](https://mlsanity.com/paper/2302.09738/) (2023)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
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- [Riemannian Nearest-Regularized Subspace Classification for Polarimetric SAR images](https://mlsanity.com/paper/2201.00337/) (2022)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)
- [Learning Log-Determinant Divergences for Positive Definite Matrices](https://mlsanity.com/paper/2104.06461/) (2021)
- [Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction](https://mlsanity.com/paper/2403.06677/) (2024)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Accurate and fast matrix factorization for low-rank learning](https://mlsanity.com/paper/2104.10785/) (2021)
- [Riemannian Complex Matrix Convolution Network for PolSAR Image Classification](https://mlsanity.com/paper/2312.03378/) (2023)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)
- [Reconciliation of Statistical and Spatial Sparsity For Robust Image and Image-Set Classification](https://mlsanity.com/paper/2106.00256/) (2021)
- [A Novel Riemannian Sparse Representation Learning Network for Polarimetric SAR Image Classification](https://mlsanity.com/paper/2502.15302/) (2025)
- [Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain](https://mlsanity.com/paper/2603.02483/) (2026)
- [Mirror Descent on Riemannian Manifolds](https://mlsanity.com/paper/2603.17527/) (2026)
- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)
- [A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds](https://mlsanity.com/paper/2608.14220/) (2026)

## Abstract (arXiv)

Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. While these matrices form an open subset of the Euclidean space of symmetric matrices, viewing them through the lens of non-Euclidean Riemannian geometry often turns out to be better suited in capturing several desirable data properties. However, formulating classical machine learning algorithms within such a geometry is often non-trivial and computationally expensive. Inspired by the great success of dictionary learning and sparse coding for vector-valued data, our goal in this paper is to represent data in the form of SPD matrices as sparse conic combinations of SPD atoms from a learned dictionary via a Riemannian geometric approach. To that end, we formulate a novel Riemannian optimization objective for dictionary learning and sparse coding in which the representation loss is characterized via the affine invariant Riemannian metric. We also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets demonstrate superior classification and retrieval performance using our approach when compared to sparse coding via alternative non-Riemannian formulations.