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Stable convergence guarantees and failure-mode characterization for NSA-Flow

Establish conditions under which the Non-negative Stiefel Approximating Flow (NSA-Flow) optimization algorithm achieves stable convergence across data regimes, and characterize failure modes in the presence of highly ill-conditioned or extremely noisy matrices to enable development of robust remedies.

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Background

The NSA-Flow framework introduces a soft-retraction optimization scheme that balances fidelity and orthogonality while enforcing non-negativity. Although empirical results show stable behavior across benchmarks, the authors note that global guarantees are challenging due to nonconvexity and the manifold constraints.

In the Limitations section, the authors explicitly state that they cannot guarantee stable convergence for all possible data and suggest that ill-conditioning or severe noise may induce failures. They further call for research to characterize these failure modes and develop robust solutions, indicating a concrete unresolved area concerning theoretical guarantees and practical robustness.

References

While the implementation seeks to minimize the sensitivity of the method to parameter choices (e.g.~optimizer, learning rate, etc), we cannot guarantee these methods will provide stable convergence for all possible data. Indeed, it is likely that highly ill-conditioned or extremely noisy data may lead to convergence issues or poor local minima. Further research is needed to characterize these failure modes and provide robust solutions.

— Non-Negative Stiefel Approximating Flow: Orthogonalish Matrix Optimization for Interpretable Embeddings (2511.06425 - Avants et al., 9 Nov 2025) in Section 4.4 (Limitations) within Discussion