Escaping Local Minima Provably in Non-convex Matrix Sensing: A Deterministic Framework via Simulated Lifting

2026-02-05

AI Summary

This work addresses the challenge posed by numerous spurious local minima in nonconvex low-rank matrix sensing problems, which often prevent gradient-based methods from converging to the global optimum. The authors propose a simulated lifting framework that deterministically escapes these local minima by simulating ascent directions in a higher-dimensional over-parameterized space and projecting them back to the original space, thereby provably decreasing the objective function value. This approach introduces the first deterministic escape mechanism that does not rely on random perturbations or heuristic estimates, combining Simulated Oracle Directions (SOD) with landscape simulation in the over-parameterized space to achieve both theoretical guarantees and significantly improved computational efficiency. Experiments demonstrate that the framework reliably converges to the global optimum with substantially lower computational overhead compared to explicit tensor over-parameterization methods.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Low-rank matrix sensing is a fundamental yet challenging nonconvex problem whose optimization landscape typically contains numerous spurious local minima, making it difficult for gradient-based optimizers to converge to the global optimum. Recent work has shown that over-parameterization via tensor lifting can convert such local minima into strict saddle points, an insight that also partially explains why massive scaling can improve generalization and performance in modern machine learning. Motivated by this observation, we propose a Simulated Oracle Direction (SOD) escape mechanism that simulates the landscape and escape direction of the over-parametrized space, without resorting to actually lifting the problem, since that would be computationally intractable. In essence, we designed a mathematical framework to project over-parametrized escape directions onto the original parameter space to guarantee a strict decrease of objective value from existing local minima. To the best of our knowledge, this represents the first deterministic framework that could escape spurious local minima with guarantee, especially without using random perturbations or heuristic estimates. Numerical experiments demonstrate that our framework reliably escapes local minima and facilitates convergence to global optima, while incurring minimal computational cost when compared to explicit tensor over-parameterization. We believe this framework has non-trivial implications for nonconvex optimization beyond matrix sensing, by showcasing how simulated over-parameterization can be leveraged to tame challenging optimization landscapes.

Problem

• non-convex optimization
• matrix sensing
• local minima
• over-parameterization
• optimization landscape

Innovation

• non-convex optimization
• matrix sensing
• over-parameterization
• simulated lifting
• deterministic escape