Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912839761920000 |
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| author | Hirai, Hiroshi |
| author_facet | Hirai, Hiroshi |
| contents | In this paper, we address the optimization problem of minimizing $Q(df_x)$ over a Hadamard manifold ${\cal M}$, where $f$ is a convex function on ${\cal M}$, $df_x$ is the differential of $f$ at $x \in {\cal M}$, and $Q$ is a function on the cotangent bundle of ${\cal M}$. This problem generalizes the problem of minimizing the gradient norm $\|\nabla f(x)\|$ over ${\cal M}$, studied by Hirai and Sakabe FOCS2024. We formulate a natural class of $Q$ in terms of convexity and invariance under parallel transports, and introduce a generalization of the gradient flow of $f$ that is expected to minimize $Q(df_x)$. For basic classes of manifolds, including the product of the manifolds of positive definite matrices, we prove that this gradient flow attains $\inf_{x\in {\cal M}} Q(df_x)$ in the limit, and yields a duality relation. This result is applied to the Kempf-Ness optimization for GL-actions on tensors, which is Euclidean convex optimization on the class of moment polytopes, known as the entanglement polytopes. This type of convex optimization arises from tensor-related subjects in theoretical computer science, such as quantum functional, $G$-stable rank, and noncommutative rank. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes Hirai, Hiroshi Optimization and Control Differential Geometry 90C25, 53C35 In this paper, we address the optimization problem of minimizing $Q(df_x)$ over a Hadamard manifold ${\cal M}$, where $f$ is a convex function on ${\cal M}$, $df_x$ is the differential of $f$ at $x \in {\cal M}$, and $Q$ is a function on the cotangent bundle of ${\cal M}$. This problem generalizes the problem of minimizing the gradient norm $\|\nabla f(x)\|$ over ${\cal M}$, studied by Hirai and Sakabe FOCS2024. We formulate a natural class of $Q$ in terms of convexity and invariance under parallel transports, and introduce a generalization of the gradient flow of $f$ that is expected to minimize $Q(df_x)$. For basic classes of manifolds, including the product of the manifolds of positive definite matrices, we prove that this gradient flow attains $\inf_{x\in {\cal M}} Q(df_x)$ in the limit, and yields a duality relation. This result is applied to the Kempf-Ness optimization for GL-actions on tensors, which is Euclidean convex optimization on the class of moment polytopes, known as the entanglement polytopes. This type of convex optimization arises from tensor-related subjects in theoretical computer science, such as quantum functional, $G$-stable rank, and noncommutative rank. |
| title | Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes |
| topic | Optimization and Control Differential Geometry 90C25, 53C35 |
| url | https://arxiv.org/abs/2511.12064 |