# Linear Classifiers in Product Space Forms

arXiv: https://arxiv.org/abs/2102.10204 · HTML: https://mlsanity.com/paper/2102.10204/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Spherical Text Embedding](https://mlsanity.com/paper/1911.01196/) (2019)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Robust Large-Margin Learning in Hyperbolic Space](https://mlsanity.com/paper/2004.05465/) (2020)
- [Hyperbolic Distance Matrices](https://mlsanity.com/paper/2005.08672/) (2020)
- [Switch Spaces: Learning Product Spaces with Sparse Gating](https://mlsanity.com/paper/2102.08688/) (2021)
- [Geometry of Similarity Comparisons](https://mlsanity.com/paper/2006.09858/) (2020)

## Built on by (most cited first, 11 of 11)

- [Latent Graph Inference using Product Manifolds](https://mlsanity.com/paper/2211.16199/) (2022)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Neural Snowflakes: Universal Latent Graph Inference via Trainable Latent Geometries](https://mlsanity.com/paper/2310.15003/) (2023)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Robust Hyperbolic Learning with Curvature-Aware Optimization](https://mlsanity.com/paper/2405.13979/) (2024)
- [HyperAid: Denoising in hyperbolic spaces for tree-fitting and hierarchical clustering](https://mlsanity.com/paper/2205.09721/) (2022)
- [Provably Accurate and Scalable Linear Classifiers in Hyperbolic Spaces](https://mlsanity.com/paper/2203.03730/) (2022)
- [Mixed-curvature decision trees and random forests](https://mlsanity.com/paper/2410.13879/) (2024)
- [Manify: A Python Library for Learning Non-Euclidean Representations](https://mlsanity.com/paper/2503.09576/) (2025)
- [Federated Classification in Hyperbolic Spaces via Secure Aggregation of Convex Hulls](https://mlsanity.com/paper/2308.06895/) (2023)
- [Mixed-Curvature Decision Trees and Random Forests](https://mlsanity.com/paper/2406.05227/) (2024)

## Abstract (arXiv)

Embedding methods for product spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Here, we address the new problem of linear classification in product space forms -- products of Euclidean, spherical, and hyperbolic spaces. First, we describe novel formulations for linear classifiers on a Riemannian manifold using geodesics and Riemannian metrics which generalize straight lines and inner products in vector spaces. Second, we prove that linear classifiers in d-dimensional space forms of any curvature have the same expressive power, i.e., they can shatter exactly d+1 points. Third, we formalize linear classifiers in product space forms, describe the first known perceptron and support vector machine classifiers for such spaces and establish rigorous convergence results for perceptrons. Moreover, we prove that the Vapnik-Chervonenkis dimension of linear classifiers in a product space form of dimension d is at least d+1. We support our theoretical findings with simulations on several datasets, including synthetic data, image data, and single-cell RNA sequencing (scRNA-seq) data. The results show that classification in low-dimensional product space forms for scRNA-seq data offers, on average, a performance improvement of ~15% when compared to that in Euclidean spaces of the same dimension.