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

arXiv: https://arxiv.org/abs/2110.13475 · HTML: https://mlsanity.com/paper/2110.13475/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 107 references that are in the map.
- First posted on arXiv 26 Oct 2021.

## Builds on (most cited first, 30 of 43)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Collaborative Filtering](https://mlsanity.com/paper/1708.05031/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Convolutional 2D Knowledge Graph Embeddings](https://mlsanity.com/paper/1707.01476/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [TuckER: Tensor Factorization for Knowledge Graph Completion](https://mlsanity.com/paper/1901.09590/) (2019)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [A Survey on Knowledge Graph-Based Recommender Systems](https://mlsanity.com/paper/2003.00911/) (2020)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Canonical Tensor Decomposition for Knowledge Base Completion](https://mlsanity.com/paper/1806.07297/) (2018)
- [Deep Learning based Recommender System: A Survey and New Perspectives](https://mlsanity.com/paper/1707.07435/) (2017)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalization](https://mlsanity.com/paper/1712.01034/) (2017)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [A neural network based on SPD manifold learning for skeleton-based hand gesture recognition](https://mlsanity.com/paper/1904.12970/) (2019)
- [Jointly Learning Explainable Rules for Recommendation with Knowledge Graph](https://mlsanity.com/paper/1903.03714/) (2019)
- [Learning Heterogeneous Knowledge Base Embeddings for Explainable Recommendation](https://mlsanity.com/paper/1805.03352/) (2018)
- [Computationally Tractable Riemannian Manifolds for Graph Embeddings](https://mlsanity.com/paper/2002.08665/) (2020)
- [Hyperbolic Representation Learning for Fast and Efficient Neural Question Answering](https://mlsanity.com/paper/1707.07847/) (2017)
- [A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classification](https://mlsanity.com/paper/2010.02053/) (2020)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)

## Built on by (most cited first, 22 of 22)

- [Heterogeneous manifolds for curvature-aware graph embedding](https://mlsanity.com/paper/2202.01185/) (2022)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)
- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis](https://mlsanity.com/paper/2203.01631/) (2022)
- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)
- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning](https://mlsanity.com/paper/2604.20308/) (2026)
- [Normed Spaces for Graph Embedding](https://mlsanity.com/paper/2312.01502/) (2023)
- [Universal kernels via harmonic analysis on Riemannian symmetric spaces](https://mlsanity.com/paper/2506.19245/) (2025)
- [Siegel Neural Networks](https://mlsanity.com/paper/2511.09577/) (2025)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [Expressivity of congruence-based architectures for DNNs on positive-definite matrices](https://mlsanity.com/paper/2606.02490/) (2026)
- [LieBN: Batch Normalization over Lie Groups](https://mlsanity.com/paper/2607.08783/) (2026)
- [Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion](https://mlsanity.com/paper/2608.16180/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)

## 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.