ManifoldFlow: SPD-Relaxed Stiefel Layers with Learnable Singular Spectrum

Abstract:Orthogonal and Stiefel layers give neural weights exact spectral control, but they also impose a strong modeling constraint: all represented singular values are fixed at one. Many settings that benefit from an orthonormal basis still need direction-dependent attenuation or amplification. We introduce ManifoldFlow, a minimal relaxation of a fixed-spectrum Stiefel layer that keeps the basis on the Stiefel manifold while learning a bounded positive spectrum through W = Q S^{1/2}, with Q^T Q = I and S positive definite. Since W^T W = S, the eigenvalues of S are exactly the squared singular values of the realized weight, making eigenvalue clipping a direct singular-value control mechanism. Across paired sequence, tabular, and image experiments, the learnable SPD spectrum improves the fixed-spectrum Stiefel counterpart in the reported settings where the Stiefel prior is useful, with the largest gains in recurrent language-model projections. Boundary cases in convolutional classifier heads clarify the intended scope: ManifoldFlow is not a universal dense-layer replacement, but a spectrum-learnable Stiefel relaxation for settings where an orthonormal basis is a useful prior. When the basis should be orthonormal, its spectrum need not be frozen. Code available at this https URL

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2607.04535 [cs.LG]

(or arXiv:2607.04535v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.04535 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.