Exploring Subgradient Methods in Hadamard Spaces

Kernkonzepte

Exploring subgradient methods in Hadamard spaces with horospherical convexity.

Zusammenfassung

The content explores subgradient methods in Hadamard spaces, focusing on horospherical convexity. It introduces a new iteration approach that does not rely on tangent spaces or exponential maps. The complexity analysis shows the applicability of the method in various spaces, including CAT(0) cubical complexes. The algorithm involves a projected subgradient iteration with projection and support oracles. Examples illustrate circumcenters, intersecting balls, and distance envelopes. The discussion contrasts traditional subgradients with the support oracle's geometric appeal. Complexity analysis provides a theoretical foundation for the method's efficiency.

Statistiken

Complexity result of O(1/√n) over n iterations.
Mean excess objective value behaves like O(1/√n).
Hidden constant deteriorates as curvature becomes more negative.

Zitate

"The iteration we propose makes no reference to tangent spaces or exponential maps."
"Our complexity result involves no lower curvature bound."
"The algorithm does not apply to every geodesically convex objective, but its range of application is still rich."

Wichtige Erkenntnisse aus

Horoballs and the subgradient method

by Adrian S. Le... um arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.15749.pdf

Tiefere Fragen

How can the support oracle enhance optimization beyond traditional subgradients?

The support oracle offers a unique advantage in optimization by focusing on horospherically convex lower level sets of objectives rather than just geodesic convexity. This approach allows for a more geometrically intuitive understanding of the problem, as it does not rely on tangent spaces and exponential maps like traditional subgradient methods. By utilizing supporting rays and horoballs, the support oracle provides a direct path towards finding optimal solutions without the need for complex local linearization techniques. This simplicity in geometric interpretation can lead to more efficient algorithms and easier extensions to various non-Euclidean spaces.

What are the limitations of relying solely on horospherical convexity for objectives?

While horospherical convexity offers an alternative perspective for optimization in Hadamard spaces, it comes with its own set of limitations. One major constraint is that not all geodesically convex objectives will exhibit horospherical convexity in their level sets. This discrepancy means that traditional subgradient methods may not always align with the requirements imposed by horospherical geometry, limiting the applicability of this approach to a specific class of objectives only.
Additionally, working solely with horospheric properties might restrict the scope of optimization problems that can be effectively solved using this method. It may overlook certain complexities present in objective functions that cannot be adequately captured or optimized through horoball-based techniques alone.

How might this alternative approach impact optimization techniques in non-Euclidean spaces?

The introduction of an alternative approach based on horosphere-related concepts could revolutionize optimization techniques in non-Euclidean spaces by offering a fresh perspective on solving complex problems efficiently. By shifting focus from tangential constructions to global geometrical properties like supporting rays and Busemann functions, this method opens up new avenues for tackling optimization challenges across diverse spatial structures beyond Riemannian manifolds.
Moreover, embracing horospherical ideas could pave the way for developing novel algorithms tailored specifically for Hadamard spaces and other non-Euclidean geometries where traditional Euclidean methods fall short. The potential applications extend to areas such as phylogenetic trees, robotics, and other fields where optimizing over curved surfaces or intricate structures is essential but challenging using conventional approaches.