Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

The Homotopy paradigm, a general principle for solving challenging problems, appears across diverse domains such as robust optimization, global optimization, polynomial root-finding, and sampling. Practical solvers for these problems typically follow a predictor-corrector (PC) structure, but rely on hand-crafted heuristics for step sizes and iteration termination, which are often suboptimal and task-specific. To address this, we unify these problems under a single framework, which enables the design of a general neural solver. Building on this unified view, we propose Neural Predictor-Corrector (NPC), which replaces hand-crafted heuristics with automatically learned policies. NPC formulates policy selection as a sequential decision-making problem and leverages reinforcement learning to automatically discover efficient strategies. To further enhance generalization, we introduce an amortized training mechanism, enabling one-time offline training for a class of problems and efficient online inference on new instances. Experiments on four representative homotopy problems demonstrate that our method generalizes effectively to unseen instances. It consistently outperforms classical and specialized baselines in efficiency while demonstrating superior stability across tasks, highlighting the value of unifying homotopy methods into a single neural framework.

Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World

Determining the vanishing points (VPs) in a Manhattan world, as a fundamental task in many 3D vision applications, consists of jointly inferring the line-VP association and locating each VP. Existing methods are, however, either sub-optimal solvers or pursuing global optimality at a significant cost of computing time. In contrast to prior works, we introduce convex relaxation techniques to solve this task for the first time. Specifically, we employ a ``soft'' association scheme, realized via a truncated multi-selection error, that allows for joint estimation of VPs' locations and line-VP associations. This approach leads to a primal problem that can be reformulated into a quadratically constrained quadratic programming (QCQP) problem, which is then relaxed into a convex semidefinite programming (SDP) problem. To solve this SDP problem efficiently, we present a globally optimal outlier-robust iterative solver (called \textbf{GlobustVP}), which independently searches for one VP and its associated lines in each iteration, treating other lines as outliers. After each independent update of all VPs, the mutual orthogonality between the three VPs in a Manhattan world is reinforced via local refinement. Extensive experiments on both synthetic and real-world data demonstrate that \textbf{GlobustVP} achieves a favorable balance between efficiency, robustness, and global optimality compared to previous works. The code is publicly available at https://github.com/WU-CVGL/GlobustVP.