Research map / Probabilistic and scientific modeling / Dynamical systems
Hyperbolic Optimizer as a Dynamical System
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives a hyperbolic (Poincaré ball) ADMM optimizer, shows its continuous-time limit is a nonlinear ODE, and studies stability of critical points via Lyapunov analysis.
Problem
Classical optimizers such as ADMM are not adapted to hyperbolic geometry, and their dynamical/asymptotic behavior in the Poincaré ball is theoretically unstudied.
Approach
Redefines ADMM in the Poincaré ball using gyrovector operations and a tangent-space (log/exp at origin) identification, derives its continuous limit as a nonlinear ODE, and uses linearization and Lyapunov functions to analyze stability of critical points.
Key results
- Theorem 3.6: the continuous limit of hyperbolic ADMM updates (t = k/rho) is a nonlinear ODE in X with Omega-term coefficients depending implicitly on X.
- Theorem 4.4: under positive-definiteness of A and gradient/positivity conditions near the critical point, the linearized hyperbolic ADMM flow's critical point is asymptotically stable (proof assumes square A).
Limitations
- No numerical experiments; the authors state numerical comparison with ADMM in gyrovector space is future work.
- Stability proof assumes n = m (square A), and the linearization neglects high-order terms.
- Hyperbolic ADMM via Mobius operations is computationally more complex than the Euclidean version.
Experimental setup
Builds on: ADMM Franca et al. 2018 ADMM continuous limit Hirsch et al. Lyapunov theorem
Closest papers by meaning
- Generalization at the Edge of Stability (NeurIPS 2026 · similarity 0.83)
- Neural Time-Reversed Generalized Riccati Equation (AAAI 2024 · similarity 0.82)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.82)
- Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction (ICML 2025 · similarity 0.82)
- How Deep Do We Need: Accelerating Training and Inference of Neural ODEs via Control Perspective (ICML 2024 · similarity 0.82)
- DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems (ICML 2024 · similarity 0.82)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.