# RMLR: Extending Multinomial Logistic Regression into General Geometries

arXiv: https://arxiv.org/abs/2409.19433 · HTML: https://mlsanity.com/paper/2409.19433/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 76 references that are in the map.
- First posted on arXiv 28 Sep 2024.

## Builds on (most cited first, 30 of 33)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Accelerating 3D Deep Learning with PyTorch3D](https://mlsanity.com/paper/2007.08501/) (2020)
- [NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis](https://mlsanity.com/paper/1604.02808/) (2016)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Domain-Specific Batch Normalization for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1906.03950/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [MOABB: Trustworthy algorithm benchmarking for BCIs](https://mlsanity.com/paper/1805.06427/) (2018)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [Deep Learning on Lie Groups for Skeleton-based Action Recognition](https://mlsanity.com/paper/1612.05877/) (2016)
- [Deep CNNs Meet Global Covariance Pooling: Better Representation and Generalization](https://mlsanity.com/paper/1904.06836/) (2019)
- [Training Deep Networks with Structured Layers by Matrix Backpropagation](https://mlsanity.com/paper/1509.07838/) (2015)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Why Approximate Matrix Square Root Outperforms Accurate SVD in Global Covariance Pooling?](https://mlsanity.com/paper/2105.02498/) (2021)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)
- [Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation](https://mlsanity.com/paper/1807.10479/) (2018)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [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)
- [On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition](https://mlsanity.com/paper/2205.13282/) (2022)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)

## Built on by (most cited first, 8 of 8)

- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning](https://mlsanity.com/paper/2605.04406/) (2026)
- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Riemannian Networks over Full-Rank Correlation Matrices](https://mlsanity.com/paper/2605.19073/) (2026)
- [Riemannian Geometry for Pre-trained Language Model Embeddings](https://mlsanity.com/paper/2607.07047/) (2026)
- [LieBN: Batch Normalization over Lie Groups](https://mlsanity.com/paper/2607.08783/) (2026)
- [Nested Inductive Bias Framework for SPD Manifold Learning](https://mlsanity.com/paper/2609.04466/) (2026)

## Abstract (arXiv)

Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning. To better classify the manifold-valued features, researchers have started extending Euclidean multinomial logistic regression (MLR) into Riemannian manifolds. However, existing approaches suffer from limited applicability due to their strong reliance on specific geometric properties. This paper proposes a framework for designing Riemannian MLR over general geometries, referred to as RMLR. Our framework only requires minimal geometric properties, thus exhibiting broad applicability and enabling its use with a wide range of geometries. Specifically, we showcase our framework on the Symmetric Positive Definite (SPD) manifold and special orthogonal group, i.e., the set of rotation matrices. On the SPD manifold, we develop five families of SPD MLRs under five types of power-deformed metrics. On rotation matrices we propose Lie MLR based on the popular bi-invariant metric. Extensive experiments on different Riemannian backbone networks validate the effectiveness of our framework.