mlsanityData to 7 Oct 2026

Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry

arXiv 2407.02607 ↗First posted 2 Jul 2024math.DG cs.LG math.MGcited by 4

Not on the map. Why?

Abstract (arXiv)

Recent advances in Symmetric Positive Definite (SPD) matrix learning show that Riemannian metrics are fundamental to effective SPD neural networks. Motivated by this, we revisit the geometry of the Cholesky factors and uncover a simple product structure that enables convenient metric design. Building on this insight, we propose two fast and stable SPD metrics, Power--Cholesky Metric (PCM) and Bures--Wasserstein--Cholesky Metric (BWCM), derived via Cholesky decomposition. Compared with existing SPD metrics, the proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability. We further apply our metrics to construct Riemannian Multinomial Logistic Regression (MLR) classifiers and residual blocks for SPD neural networks. Experiments on SPD deep learning, numerical stability analyses, and tensor interpolation demonstrate the effectiveness, efficiency, and robustness of our metrics. The code is available at https://github.com/GitZH-Chen/PCM_BWCM.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 26 of its 61 references that are in the map. First posted on arXiv 2 Jul 2024.

Builds on

26 of 61 references.

References · 26

  1. 2018Hyperbolic Neural Networkscited by 393
  2. 2023Flow Matching on General Geometriescited by 260
  3. 2017First-Person Hand Action Benchmark with RGB-D Videos and 3D Hand Pose Annotationscited by 229
  4. 2018Riemannian Adaptive Optimization Methodscited by 187
  5. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  6. 2019Mixed-curvature Variational Autoencoderscited by 85
  7. 2019Trivializations for Gradient-Based Optimization on Manifoldscited by 78
  8. 2019Riemannian batch normalization for SPD neural networkscited by 72
  9. 2017Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalizationcited by 65
  10. 2022SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEGcited by 53
  11. 2022MAtt: A Manifold Attention Network for EEG Decodingcited by 43
  12. 2016Deep Learning on Lie Groups for Skeleton-based Action Recognitioncited by 42
14 more
  1. 2019Deep CNNs Meet Global Covariance Pooling: Better Representation and Generalizationcited by 42
  2. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  3. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  4. 2023Riemannian Residual Neural Networkscited by 21
  5. 2018A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matricescited by 18
  6. 2023Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approachcited by 18
  7. 2022Fast Differentiable Matrix Square Root and Inverse Square Rootcited by 16
  8. 2023Exploring Data Geometry for Continual Learningcited by 14
  9. 2023Adaptive Log-Euclidean Metrics for SPD Matrix Learningcited by 13
  10. 2023SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Spacecited by 13
  11. 2021Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometrycited by 10
  12. 2020ManifoldNorm: Extending normalizations on Riemannian Manifoldscited by 8
  13. 2024A Lie Group Approach to Riemannian Batch Normalizationcited by 8
  14. 2024Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for Riemannian Manifoldcited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 0; 2026: 2.

4 papers.

  1. 2024Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometrycited by 9
  2. 2024RMLR: Extending Multinomial Logistic Regression into General Geometriescited by 8
  3. 2026Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learningcited by 1
  4. 2026Riemannian Networks over Full-Rank Correlation Matricescited by 0