最佳逼近随机投影神经网络：收敛理论与实际应用
Random Projection Neural Networks of Best Approximation: Convergence theory and practical applications
AI总结：
本文研究了前馈神经网络的最佳逼近性质，提出具有固定内部权重和偏置的随机投影神经网络（RPNNs），证明其在逼近无限可微函数时具有指数收敛率，并在五个基准问题上表现出与Legendre多项式相当的性能。
摘要：
我们研究了前馈神经网络（FNN）的最佳逼近概念，并通过随机投影神经网络（RPNNs）的视角探讨了其收敛性质。RPNNs具有一劳永逸地预先确定的内部权重和偏置，提供了计算效率。我们证明，对于任何此类RPNN家族，在采用非多项式无限可微激活函数时，存在一种外部权重选择，在逼近任何无限可微函数时表现出指数收敛率。为了说明，我们在五个基准函数逼近问题上测试了所提出的基于RPNN的函数逼近方法，并精简地选择了基函数。结果表明，RPNNs实现了与Legendre多项式等成熟方法相当的性能，突显了其在高效且精确的函数逼近方面的潜力。
英文摘要：
We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal weights and biases, offering computational efficiency. We demonstrate that there exists a choice of external weights, for any family of such RPNNs, with non-polynomial infinitely differentiable activation functions, that exhibit an exponential convergence rate when approximating any infinitely differentiable function. For illustration purposes, we test the proposed RPNN-based function approximation, with parsimoniously chosen basis functions, across five benchmark function approximation problems. Results show that RPNNs achieve comparable performance to established methods such as Legendre Polynomials, highlighting their potential for efficient and accurate function approximation.
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