Learning with Group Invariant Features: A Kernel Perspective
This work addresses the challenge of incorporating group invariance into machine learning models, offering a theoretical framework that bridges invariant features with kernel methods, though it appears incremental in nature.
The paper tackles the problem of learning group-invariant features by introducing a random feature map based on I-theory, which approximates an invariant kernel and reduces sample complexity for signal classification, with quantified error rates in empirical risk minimization.
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.