# McTorch, a manifold optimization library for deep learning

arXiv: https://arxiv.org/abs/1810.01811 · HTML: https://mlsanity.com/paper/1810.01811/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Oct 2018.

## Built on by (most cited first, 16 of 16)

- [Tangent Space Backpropagation for 3D Transformation Groups](https://mlsanity.com/paper/2103.12032/) (2021)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [SECA: Semantically Equivalent and Coherent Attacks for Eliciting LLM Hallucinations](https://mlsanity.com/paper/2510.04398/) (2025)
- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [NCVX: A General-Purpose Optimization Solver for Constrained Machine and Deep Learning](https://mlsanity.com/paper/2210.00973/) (2022)
- [tensorflow-riemopt: A Library for Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2105.13921/) (2021)
- [Coordinate descent on the orthogonal group for recurrent neural network training](https://mlsanity.com/paper/2108.00051/) (2021)
- [NCVX: A User-Friendly and Scalable Package for Nonconvex Optimization in Machine Learning](https://mlsanity.com/paper/2111.13984/) (2021)
- [Riemannian Optimization on Relaxed Indicator Matrix Manifold](https://mlsanity.com/paper/2503.20505/) (2025)
- [MCNC: Manifold-Constrained Reparameterization for Neural Compression](https://mlsanity.com/paper/2406.19301/) (2024)
- [Parallelised Differentiable Straightest Geodesics for 3D Meshes](https://mlsanity.com/paper/2603.15780/) (2026)
- [Rieoptax: Riemannian Optimization in JAX](https://mlsanity.com/paper/2210.04840/) (2022)
- [TAOTF: A Two-stage Approximately Orthogonal Training Framework in Deep Neural Networks](https://mlsanity.com/paper/2211.13902/) (2022)

## Abstract (arXiv)

In this paper, we introduce McTorch, a manifold optimization library for deep learning that extends PyTorch. It aims to lower the barrier for users wishing to use manifold constraints in deep learning applications, i.e., when the parameters are constrained to lie on a manifold. Such constraints include the popular orthogonality and rank constraints, and have been recently used in a number of applications in deep learning. McTorch follows PyTorch's architecture and decouples manifold definitions and optimizers, i.e., once a new manifold is added it can be used with any existing optimizer and vice-versa. McTorch is available at https://github.com/mctorch .