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Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes

Hiroshi Hirai

TL;DR

The paper develops a generalized gradient-flow framework on Hadamard manifolds by introducing a convex, parallel-transport invariant function $Q$ and the associated $Q$-gradient flow, establishing a duality with the recession function $f^{\infty}$ and proving convergence to the infimum of $Q(df_x)$ for several Hadamard classes. It then applies this theory to Kempf-Ness optimization for GL-actions on tensors, recasting convex optimization on moment polytopes as Euclidean convex optimization on entanglement polytopes and deriving a duality formula involving $\Phi_v^{\infty}$ and $S^*$; the flow yields accumulation points that minimize the objective on the entanglement polytope. The framework connects to concrete tensor-analytic objects, including quantum functionals $F_{\theta}$, the $G$-stable rank, and noncommutative rank, providing dual characterizations and suggesting algorithmic paths (via $Q$-gradient or $S$-gradient flows) for these problems. This work thus bridges geometric analysis on Hadamard manifolds with invariant convex optimization on moment polytopes, enabling potential NP $\cap$ co-NP insights for entanglement- polytope problems and related tensor-rank questions.

Abstract

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.

Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes

TL;DR

The paper develops a generalized gradient-flow framework on Hadamard manifolds by introducing a convex, parallel-transport invariant function and the associated -gradient flow, establishing a duality with the recession function and proving convergence to the infimum of for several Hadamard classes. It then applies this theory to Kempf-Ness optimization for GL-actions on tensors, recasting convex optimization on moment polytopes as Euclidean convex optimization on entanglement polytopes and deriving a duality formula involving and ; the flow yields accumulation points that minimize the objective on the entanglement polytope. The framework connects to concrete tensor-analytic objects, including quantum functionals , the -stable rank, and noncommutative rank, providing dual characterizations and suggesting algorithmic paths (via -gradient or -gradient flows) for these problems. This work thus bridges geometric analysis on Hadamard manifolds with invariant convex optimization on moment polytopes, enabling potential NP co-NP insights for entanglement- polytope problems and related tensor-rank questions.

Abstract

In this paper, we address the optimization problem of minimizing over a Hadamard manifold , where is a convex function on , is the differential of at , and is a function on the cotangent bundle of . This problem generalizes the problem of minimizing the gradient norm over , studied by Hirai and Sakabe FOCS2024. We formulate a natural class of in terms of convexity and invariance under parallel transports, and introduce a generalization of the gradient flow of that is expected to minimize . For basic classes of manifolds, including the product of the manifolds of positive definite matrices, we prove that this gradient flow attains 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, -stable rank, and noncommutative rank.

Paper Structure

This paper contains 15 sections, 24 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hadamard manifold
  4. Convex analysis (on Hermitian matrices)
  5. Manifold ${\cal P}_n$ of positive definite matrices
  6. Euclidean building $C{\cal P}_n^{\infty}$.
  7. $Q$-gradient flow and asymptotic duality
  8. $Q$-subgradient method.
  9. Convex optimization on entanglement polytopes
  10. GL-action on tensors and entanglement polytopes
  11. Quantum functional
  12. Seeing that quantum functionals are spectral points.
  13. $G$-stable rank
  14. Noncommutative rank
  15. Proof of Lemma \ref{['lem:ui-convex=>g-convex']}

Key Result

Lemma 2.3

In this identification ${\cal H}_n = {\cal B}_n$, any unitarily invariant convex function $f:{\cal H}_n \to \overline{{\mathbb{R}}}$ is geodesically convex on ${\cal B}_n$.

Theorems & Definitions (50)

  • Example 2.1: Unitarily invariant norms $\|\cdot\|_{\rm op}$ and $\|\cdot \|_{\rm tr}$
  • Example 2.2: von Neumann entropy
  • Lemma 2.3
  • Example 3.1: Euclidean space
  • Example 3.2: Dual norm
  • Example 3.3: PD-manifold and unitarily invariant convex function
  • Lemma 3.4: Weak duality
  • proof
  • Example 3.5
  • Definition 3.6: $Q$-gradient flow
  • ...and 40 more