Manifold constrained steepest descent

• URL: http://arxiv.org/abs/2601.21487v1
• Date: Thu, 29 Jan 2026 10:08:37 GMT
• Title: Manifold constrained steepest descent
• Authors: Kaiwei Yang, Lexiao Lai,
• Abstract summary: We propose emphManifold Constrained Steepest Descent (MCSD), a single-loop framework for optimization over manifold.<n>We also introduce emphSPEL, the spectral specialization of MCSD on the Stiefel manifold.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Norm-constrained linear minimization oracle (LMO)-based optimizers such as spectral gradient descent and Muon are attractive in large-scale learning, but extending them to manifold-constrained problems is nontrivial and often leads to nested-loop schemes that solve tangent-space subproblems iteratively. We propose \emph{Manifold Constrained Steepest Descent} (MCSD), a single-loop framework for optimization over manifolds that selects a norm-induced steepest-descent direction via an LMO applied to the Riemannian gradient, and then returns to the manifold via projection. Under standard smoothness assumptions, we establish convergence guarantees for MCSD and a stochastic momentum variant. We further introduce \emph{SPEL}, the spectral-norm specialization of MCSD on the Stiefel manifold, which admits scalable implementations via fast matrix sign computations. Experiments on PCA, orthogonality-constrained CNNs, and manifold-constrained LLM adapter tuning demonstrate improved stability and competitive performance relative to standard Riemannian baselines and existing manifold-aware LMO methods.

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