Optimization on Manifolds
Abstract
This paper presents a comprehensive framework for optimization on Riemannian manifolds with applications to machine learning. We develop novel convergence guarantees and demonstrate superior performance on constrained optimization problems arising in neural networks and dimensionality reduction.
Cite this work
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Dr. Smith, SSL Team (2025). Optimization on Manifolds. Journal of Mathematical Optimization. https://doi.org/10.1000/xyz123@inproceedings{manifold_optimization_2025,
title={Optimization on Manifolds},
author={Dr. Smith, SSL Team},
year={2025},
booktitle={Journal of Mathematical Optimization},
volume={42},
number={3},
pages={123-145},
doi={10.1000/xyz123},
}Why it matters to SSL
Theoretical foundation for Fogo OS algorithms and optimization routines used in our urban modeling platform.
Why it matters to SSL: Theoretical foundation for Fogo OS algorithms.
Summary
Novel geometric optimization methods that respect manifold structure
Rigorous convergence analysis with improved rates over Euclidean methods
Applications to neural networks, dimensionality reduction, and constrained ML
Technical Contributions
Our main contributions include:
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Geometric Convergence Analysis: We prove that our Riemannian gradient methods achieve convergence rates under geodesic convexity assumptions.
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Adaptive Step Size Selection: A novel adaptive step size rule that respects the manifold geometry while maintaining global convergence guarantees.
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Practical Algorithms: Efficient implementations for common manifolds including the Stiefel manifold, positive definite matrices, and hyperbolic spaces.
Experimental Results
We evaluate our methods on several machine learning tasks:
- Matrix Completion: 15% improvement in RMSE over standard methods
- Dimensionality Reduction: Better preservation of local neighborhood structure
- Neural Network Training: Faster convergence with fewer parameters
Notes
This fundamental mathematical research directly supports our platform development by providing principled optimization methods that can handle the geometric constraints inherent in urban modeling and machine learning applications. The theoretical guarantees ensure reliability in production systems.
Implementation
The algorithms described in this paper are implemented in our open-source optimization library and integrated into Fogo OS for real-time urban data processing.