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High curvature means low-rank: On the sectional curvature of Grassmann and Stiefel manifolds and the underlying matrix trace inequalities (2403.01879v3)

Published 4 Mar 2024 in math.NA and cs.NA

Abstract: Methods and algorithms that work with data on nonlinear manifolds are collectively summarized under the term Riemannian computing'. In practice, curvature can be a key limiting factor for the performance of Riemannian computing methods. Yet, curvature can also be a powerful tool in the theoretical analysis of Riemannian algorithms. In this work, we investigate the sectional curvature of the Stiefel and Grassmann manifold. On the Grassmannian, tight curvature bounds are known since the late 1960ies. On the Stiefel manifold under the canonical metric, it was believed that the sectional curvature does not exceed 5/4. Under the Euclidean metric, the maximum was conjectured to be at 1. For both manifolds, the sectional curvature is given by the Frobenius norm of certain structured commutator brackets of skew-symmetric matrices. We provide refined inequalities for such terms and pay special attention to the maximizers of the curvature bounds. In this way, we prove for the Stiefel manifold that the global bounds of 5/4 (canonical metric) and 1 (Euclidean metric) hold indeed. With this addition, a complete account of the curvature bounds in all admissible dimensions is obtained. We observe thathigh curvature means low-rank', more precisely, for the Stiefel and Grassmann manifolds under the canonical metric, the global curvature maximum is attained at tangent plane sections that are spanned by rank-two matrices, while the extreme curvature cases of the Euclidean Stiefel manifold occur for rank-one matrices. Numerical examples are included for illustration purposes.

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Citations (5)
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Summary

  • The paper confirms the global curvature bound of 5/4 for the Stiefel manifold using matrix norm inequalities and quotient space analysis.
  • The study reveals that maximal curvature occurs only in sections spanned by low-rank matrices, linking curvature behavior with matrix rank.
  • The established bounds enhance theoretical understanding and guide Riemannian computing methods in optimization and machine learning.

Sectional Curvature Bounds on Stiefel and Grassmann Manifolds

Introduction

Explorations into Riemannian geometry have unveiled significant insights into the behavior and properties of data residing on non-linear spaces. Among these findings, curvature plays a pivotal role, influencing both the theoretical foundations and practical applications of Riemannian computing. In recent studies, the sectional curvature of the Stiefel and Grassmann manifolds has been scrutinized, revealing intricate relationships between curvature bounds, matrix inequalities, and the structural aspects of these manifolds.

Curvature in Riemannian Geometry

The sectional curvature of a manifold quantifies the degree to which the geometry of the manifold differs from flat Euclidean space within a given plane of directions. For the Stiefel and Grassmann manifolds, curvature intricacies arise due to their quotient space structures, leading to distinct curvature formulations. Historically, the curvature of the Grassmannian has been well-characterized, with established bounds dating back to the late 1960s. However, the curvature bounds for the Stiefel manifold, particularly under the canonical metric, have been less clear, with previous studies suggesting an upper limit of 5/4 without definitive proof for general dimensions.

Insights and Contributions

This research addresses the gap in understanding related to the Stiefel manifold's sectional curvature. Through refined matrix inequalities and a meticulous analysis of curvature maximizers, this paper confirms the previously conjectured global bound of 5/4 for the Stiefel manifold under the canonical metric.

  • Key Results and Methodology: By examining the curvature from the quotient space viewpoint and employing advanced matrix norm inequalities, the paper establishes firm bounds for the sectional curvature across all admissible dimensions for the Stiefel and Grassmann manifolds. Notably, it is proven that the global curvature maximum is achieved only by sections spanned by low-rank matrices, underscoring the principle that 'high curvature implies low rank'.
  • Theoretical Implications: Beyond settling the conjecture about the Stiefel manifold's curvature bounds, the findings enrich our understanding of the geometric behavior of these manifolds. Importantly, the observation that maximum curvature corresponds to low-rank matrix sections provides a deeper insight into the structure of the manifold's tangent spaces.
  • Practical Relevance: The refined curvature bounds have direct implications for the performance and analysis of Riemannian computing methods. In areas such as optimization, machine learning, and data science, where algorithms often operate on or within these manifolds, a precise understanding of curvature can guide the development of more efficient computational tools and theoretical models.

Future Directions

The confirmation of curvature bounds on the Stiefel and Grassmann manifolds opens several avenues for further research. Investigating how these findings can be leveraged to improve algorithmic strategies in Riemannian computing stands out as a promising direction. Additionally, exploring the curvature properties of other related manifolds could yield further insights into the geometric foundations of non-linear data structures and computational spaces.

Conclusion

By addressing the longstanding conjecture regarding the sectional curvature of the Stiefel manifold and elucidating the relationship between curvature and matrix rank, this work significantly advances our comprehension of Riemannian manifolds' geometric properties. These findings not only reinforce the theoretical underpinnings of Riemannian geometry but also have practical implications for its application across various scientific and engineering disciplines.

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