证明了组合式逼近可严格优于叠加式逼近，打破经典认知。
Compositional Approximation Can Strictly Outperform Superpositional Approximation
- 用结构化函数类对比两种逼近方式的性能
- 构造实例显示两者误差率差距可任意大
- 适合研究逼近理论与神经网络优势的学者
许多经典函数类的最优逼近依赖于叠加方法，即通过字典元素的线性组合构建近似函数。这里的最优指：当参数数量增加时，一致逼近误差以最高可能的多项式速率下降，且参数可编码为长度与参数量成比例（对数因子内）的比特串。尽管组合式方法（如神经网络）结构不同，但通过约束保证此类编码后，其逼近性能可与叠加方法相当。本文研究一类具有特定结构特性的函数类，其叠加逼近率被严格限制在低于组合逼近率。特别地，我们构造出任意大误差差距的显式例子。
原文摘要 · Abstract (English)
Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.
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