mlsanityData to 7 Oct 2026

Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices

arXiv 2110.13475 ↗First posted 26 Oct 2021cs.LG cs.CGcited by 22

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Abstract (arXiv)

We propose the use of the vector-valued distance to compute distances and extract geometric information from the manifold of symmetric positive definite matrices (SPD), and develop gyrovector calculus, constructing analogs of vector space operations in this curved space. We implement these operations and showcase their versatility in the tasks of knowledge graph completion, item recommendation, and question answering. In experiments, the SPD models outperform their equivalents in Euclidean and hyperbolic space. The vector-valued distance allows us to visualize embeddings, showing that the models learn to disentangle representations of positive samples from negative ones.

Key facts. Cited by 22 papers in the map (data through 7 Oct 2026). Builds on 43 of its 107 references that are in the map. First posted on arXiv 26 Oct 2021.

Builds on

43 of 107 references.

References · 43

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  12. 2018Riemannian Adaptive Optimization Methodscited by 187
31 more
  1. 2019Quaternion Knowledge Graph Embeddingscited by 175
  2. 2020A Survey on Knowledge Graph-Based Recommender Systemscited by 170
  3. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  4. 2018Canonical Tensor Decomposition for Knowledge Base Completioncited by 150
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  6. 2019Multi-relational Poincar'e Graph Embeddingscited by 126
  7. 2013Manopt, a Matlab toolbox for optimization on manifoldscited by 112
  8. 2019Constant Curvature Graph Convolutional Networkscited by 106
  9. 2019Riemannian batch normalization for SPD neural networkscited by 72
  10. 2017Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalizationcited by 65
  11. 2014Geodesic Exponential Kernels: When Curvature and Linearity Conflictcited by 47
  12. 2019A neural network based on SPD manifold learning for skeleton-based hand gesture recognitioncited by 46
  13. 2019Jointly Learning Explainable Rules for Recommendation with Knowledge Graphcited by 40
  14. 2018Learning Heterogeneous Knowledge Base Embeddings for Explainable Recommendationcited by 39
  15. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  16. 2017Hyperbolic Representation Learning for Fast and Efficient Neural Question Answeringcited by 23
  17. 2020A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classificationcited by 23
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  19. 2014From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matricescited by 22
  20. 2014Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matricescited by 20
  21. 2018A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matricescited by 18
  22. 2016Kernel Sparse Subspace Clustering on Symmetric Positive Definite Manifoldscited by 10
  23. 2020Generalized Nonlinear and Finsler Geometry for Roboticscited by 10
  24. 2017Deep manifold-to-manifold transforming network for action recognitioncited by 9
  25. 2015Riemannian Metric Learning for Symmetric Positive Definite Matricescited by 8
  26. 2017Learning a Robust Representation via a Deep Network on Symmetric Positive Definite Manifoldscited by 7
  27. 2017Clustering in Hilbert simplex geometrycited by 4
  28. 2018Convex Class Model on Symmetric Positive Definite Manifoldscited by 3
  29. 2021Augmenting the User-Item Graph with Textual Similarity Modelscited by 3
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 6; 2024: 4; 2025: 3; 2026: 7.

22 papers.

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  1. 2022Heterogeneous manifolds for curvature-aware graph embeddingcited by 23
  2. 2023Riemannian Residual Neural Networkscited by 21
  3. 2023Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approachcited by 18
  4. 2022Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysiscited by 14
  5. 2023Adaptive Log-Euclidean Metrics for SPD Matrix Learningcited by 13
  6. 2024Matrix Manifold Neural Networks++cited by 13
  7. 2023Riemannian Multinomial Logistics Regression for SPD Neural Networkscited by 11
  8. 2023Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matricescited by 10
  9. 2024RMLR: Extending Multinomial Logistic Regression into General Geometriescited by 8
  10. 2024A Riemannian Framework for Learning Reduced-order Lagrangian Dynamicscited by 7
  11. 2025SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challengescited by 5
  12. 2026Neural Networks on Symmetric Spaces of Noncompact Typecited by 5
10 more
  1. 2024Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometrycited by 4
  2. 2023Normed Spaces for Graph Embeddingcited by 2
  3. 2025Universal kernels via harmonic analysis on Riemannian symmetric spacescited by 2
  4. 2026Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learningcited by 2
  5. 2025Siegel Neural Networkscited by 1
  6. 2026Geometry-Preserving Neural Architectures on Manifolds with Boundarycited by 0
  7. 2026Expressivity of congruence-based architectures for DNNs on positive-definite matricescited by 0
  8. 2026LieBN: Batch Normalization over Lie Groupscited by 0
  9. 2026Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterioncited by 0
  10. 2026Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifoldcited by 0