# ManifoldFlow: Spectrum-Learnable Stiefel Optimization via SPD Relaxation

arXiv: https://arxiv.org/abs/2607.04535 · HTML: https://mlsanity.com/paper/2607.04535/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 59 references that are in the map.
- First posted on arXiv 5 Jul 2026.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
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- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)
- [All you need is a good init](https://mlsanity.com/paper/1511.06422/) (2015)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks](https://mlsanity.com/paper/1709.06079/) (2017)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)
- [Orthogonal Recurrent Neural Networks with Scaled Cayley Transform](https://mlsanity.com/paper/1707.09520/) (2017)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Scalable Optimization in the Modular Norm](https://mlsanity.com/paper/2405.14813/) (2024)
- [Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections](https://mlsanity.com/paper/1612.00188/) (2016)
- [Gated Orthogonal Recurrent Units: On Learning to Forget](https://mlsanity.com/paper/1706.02761/) (2017)
- [Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds](https://mlsanity.com/paper/2605.09238/) (2026)

## Abstract (arXiv)

Orthogonal and Stiefel constraints are widely used in neural network optimization to control weight geometry, improve conditioning, and stabilize signal propagation. Yet standard Stiefel optimization imposes a hidden restriction: while learning the directions of a linear map, it fixes every singular value to one, preventing the model from learning how strongly different directions should be amplified or suppressed. We introduce ManifoldFlow, a spectrum-learnable extension of Stiefel optimization that separates directional geometry from spectral scaling through W=QS1/2, where Q^top Q=I and Ssucc0. Beyond simply making the spectrum trainable, ManifoldFlow uses the gradient component rejected by the Stiefel constraint as a signal for how the spectrum should change, accumulates this signal across aligned tangent spaces, and jointly updates the orthonormal basis and a bounded SPD spectrum. This construction preserves exact spectral interpretability because W^top W=S, while recovering standard Stiefel optimization when S=I. We establish the geometry of the rejected gradient, characterize optimization over the resulting Stiefel--SPD product space, and derive how the learned spectrum affects conditioning, signal propagation, and spectral complexity. Across diverse tasks, ManifoldFlow consistently outperforms matched fixed-spectrum Stiefel baselines, with particularly strong gains in recurrent language modeling. Our code is available at https://github.com/Hik289/manifoldflow.git.