We analyze in this paper a random feature map based on a theory of invariance
I-theory introduced recently. More specifically, a group invariant signal
signature is obtained through cumulative distributions of group transformed
random projections. Our analysis bridges invariant feature learning with kernel
methods, as we show that this feature map defines an expected Haar integration
kernel that is invariant to the specified group action. We show how this
non-linear random feature map approximates this group invariant kernel
uniformly on a set of $N$ points. Moreover, we show that it defines a function
space that is dense in the equivalent Invariant Reproducing Kernel Hilbert
Space. Finally, we quantify error rates of the convergence of the empirical
risk minimization, as well as the reduction in the sample complexity of a
learning algorithm using such an invariant representation for signal
classification, in a classical supervised learning setting.

本文研究基于最近引入的不变理论 I 理论的随机特征映射。我们通过组变换的累积分布得到了一组不变的信号标记。本文将不变特征学习与内核方法相结合，并表明该特征映射定义了一个预期的 Haar 积分内核，对指定的组操作具有不变性。此外，本文还分析了此非线性随机特征映射如何在 N 个点上均匀近似组不变内核，并证明了它在等价的不变再生核希尔伯特空间中定义了一个密集的函数空间。最后，我们量化了在经典监督学习环境中使用这种不变信号表示进行信号分类的经验风险最小化的收敛误差率以及样本复杂度的降低。