This paper presents a method for finding a sparse representation of Barron functions. Specifically, given an $L^2$ function $f$, the inverse scale space flow is used to find a sparse measure $\mu$ minimising the $L^2$ loss between the Barron function associated to the measure $\mu$ and the function $f$. The convergence properties of this method are analysed in an ideal setting and in the cases of measurement noise and sampling bias. In an ideal setting the objective decreases strictly monotone in time to a minimizer with $\mathcal{O}(1/t)$, and in the case of measurement noise or sampling bias the optimum is achieved up to a multiplicative or additive constant. This convergence is preserved on discretization of the parameter space, and the minimizers on increasingly fine discretizations converge to the optimum on the full parameter space.

翻译：本文提出了一种用于学习Barron函数稀疏表示的方法。具体而言，给定一个$L^2$函数$f$，利用逆尺度空间流寻找一个稀疏测度$\mu$，使得与该测度$\mu$相关联的Barron函数与目标函数$f$之间的$L^2$损失最小化。本文在理想环境、测量噪声及采样偏差情形下分析了该方法的收敛特性。在理想环境中，目标函数随时间严格单调递减至极小值点，收敛速度为$\mathcal{O}(1/t)$；在存在测量噪声或采样偏差时，最优解可在乘法或加法常数误差范围内达到。该收敛性在参数空间离散化后仍得以保持，且随着离散化网格逐渐加密，对应的极小值点收敛至全参数空间上的最优解。