让随机特征模型的激活函数可学习，提升表达能力与效率

The random feature (RF) method is a powerful kernel approximation technique, but it typically uses fixed activation functions, limiting its adaptability across diverse tasks. To overcome this limitation, we introduce the Random Feature Model with Learnable Activation Functions (RFLAF), a novel statistical model that parameterizes activation functions as weighted sums of basis functions within the random feature framework. Examples of basis functions include radial basis functions (RBFs), spline functions, polynomials, and so forth. For theoretical results, we consider RBFs as representative basis functions. We start with a single RBF as the activation, and then extend the results to multiple RBFs, demonstrating that RF models with a learnable activation component substantially expand the represented function space. We provide estimates on the required number of samples and random features to achieve low excess risk. In our experiments, we test RFLAF with three types of bases: radial basis functions, spline functions and polynomials. Experimental results show that RFLAFs with RBFs and splines consistently outperform other RF models, where RBFs are three times more computationally efficient than splines. We then unfreeze the first-layer parameters and retrain the models, validating the expressivity advantage of learnable activation components on regular two-layer neural networks. Our work provides a deeper understanding of learnable activation components within modern neural network architectures.