# Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition

arXiv: https://arxiv.org/abs/1608.04914 · HTML: https://mlsanity.com/paper/1608.04914/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2016.

## Built on by (most cited first, 9 of 9)

- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [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)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [Beyond First-Order: Learning Riemannian Geometries for Invariant Visual Place Recognition](https://mlsanity.com/paper/2602.00841/) (2026)
- [Geometric sparsification in recurrent neural networks](https://mlsanity.com/paper/2406.06290/) (2024)
- [Learning Chebyshev Basis in Graph Convolutional Networks for Skeleton-based Action Recognition](https://mlsanity.com/paper/2104.05482/) (2021)
- [K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups](https://mlsanity.com/paper/2603.19601/) (2026)

## Abstract (arXiv)

Symmetric Positive Definite (SPD) matrices have been widely used for data representation in many visual recognition tasks. The success mainly attributes to learning discriminative SPD matrices with encoding the Riemannian geometry of the underlying SPD manifold. In this paper, we propose a geometry-aware SPD similarity learning (SPDSL) framework to learn discriminative SPD features by directly pursuing manifold-manifold transformation matrix of column full-rank. Specifically, by exploiting the Riemannian geometry of the manifold of fixed-rank Positive Semidefinite (PSD) matrices, we present a new solution to reduce optimizing over the space of column full-rank transformation matrices to optimizing on the PSD manifold which has a well-established Riemannian structure. Under this solution, we exploit a new supervised SPD similarity learning technique to learn the transformation by regressing the similarities of selected SPD data pairs to their ground-truth similarities on the target SPD manifold. To optimize the proposed objective function, we further derive an algorithm on the PSD manifold. Evaluations on three visual classification tasks show the advantages of the proposed approach over the existing SPD-based discriminant learning methods.