Основні поняття
Sparsifying the magnetic Laplacian using multi-type spanning forests for spectral approximations.
Анотація
The content discusses sparsification of the regularized magnetic Laplacian using multi-type spanning forests. It explores applications in angular synchronization and semi-supervised learning, providing statistical guarantees and practical implications. The paper introduces a novel approach, sparsify-and-eigensolve, for approximating eigenvectors and sparsify-and-precondition for improving numerical convergence. Sampling methods like CyclePopping are highlighted for fast sampling of MTSFs. Empirical results on ranking and preconditioning systems are presented, along with limitations and notations used.
- Introduction:
- Defines U(1)-connection graph with complex phases.
- Related Work:
- Discusses Laplacian sparsification methods and spectral approaches.
- The Magnetic Laplacian:
- Explains the spectral approach to angular synchronization.
- Multi-Type Spanning Forests:
- Introduces determinantal point processes favoring inconsistent cycles.
- Statistical Guarantees:
- Provides guarantees for sparsification with MTSFs.
- Matrix Chernoff Bound:
- Presents a bound with intrinsic dimension for DPPs.
- Sampling Methods:
- Details CyclePopping algorithm for weakly inconsistent graphs.
- Practical Applications:
- Illustrates empirical results on ranking and semi-supervised learning.