Topology-optimized tensor decomposition: A global convergence proof for non-convex algorithms

Abstract

This paper proposes a Topological Optimization Homotopy Framework (TOHF) and provide the first rigorous proof of its global convergence to solve the non-convex optimization challenges in high-dimensional tensor decomposition. TOHF transforms the tensor algebraic structure into topological invariants to construct two-layer manifold optimization paths with global guidance. By integrating homotopy parameter scheduling and curvature correction, it overcomes the saddle-dominated energy landscape. Theoretically, TOHF achieves global convergence under the Wasserstein topological distance. Experiments in recommender systems, medical imaging, and neural network compression show that TOHF provides significant improvements in accuracy and convergence efficiency compared to state-of-the-art methods, effectively mitigating local minima traps, accelerating objective function descent, and enhancing solution set stability. This study establishes a topology-based convergence analysis paradigm for high-dimensional nonconvex optimization problems, which is of theoretical guidance for large-scale data processing applications.