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Exploring the non-convexity in machine learning using quantum-inspired optimization

The escalating complexity of modern machine learning necessitates solving challenging non-convex optimization problems, particularly in high-dimensional regimes and scenarios contaminated by gross outliers. Traditional approaches, relying on convex relaxations or specialized local search heuristics, frequently succumb to suboptimal local minima and fail to recover the true underlying discrete structures. In this paper, we propose treating these non-convex challenges as a global search problem and introduce a unified framework based on Quantum-Inspired Evolutionary Optimization (QIEO). By leveraging a probabilistic representation inspired by quantum superposition, QIEO maintains a global view of the search space, enabling it to tunnel through local optima that trap conventional gradient-based and greedy solvers. We comprehensively evaluate QIEO across diverse non-convex applications, including sparse signal recovery (gene expression analysis and compressed sensing) and robust linear regression. Extensive benchmarking against state-of-the-art continuous solvers (ADAM, Differential Evolution), classical metaheuristics (Genetic Algorithms), and specialized non-convex algorithms (Iterative Hard Thresholding) demonstrates that QIEO consistently achieves superior structural fidelity, lower mean squared error, and enhanced robustness without support inflation. Our findings suggest that embracing a quantum-inspired global search provides a resilient, unified paradigm for overcoming the inherent intractability of discrete nonconvex machine learning landscapes.

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