# Learning with Group Invariant Features: A Kernel Perspective

arXiv: https://arxiv.org/abs/1506.02544 · HTML: https://mlsanity.com/paper/1506.02544/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2015.

## Built on by (most cited first, 13 of 13)

- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Provably Strict Generalisation Benefit for Invariance in Kernel Methods](https://mlsanity.com/paper/2106.02346/) (2021)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [The good, the bad and the ugly sides of data augmentation: An implicit spectral regularization perspective](https://mlsanity.com/paper/2210.05021/) (2022)
- [Understanding the Generalization Benefit of Model Invariance from a Data Perspective](https://mlsanity.com/paper/2111.05529/) (2021)
- [Understanding Augmentation-based Self-Supervised Representation Learning via RKHS Approximation and Regression](https://mlsanity.com/paper/2306.00788/) (2023)
- [Emergent Equivariance in Deep Ensembles](https://mlsanity.com/paper/2403.03103/) (2024)
- [Debiasing Conditional Stochastic Optimization](https://mlsanity.com/paper/2304.10613/) (2023)
- [Rectifying Shortcut Behaviors in Preference-based Reward Learning](https://mlsanity.com/paper/2510.19050/) (2025)
- [Federated Conditional Stochastic Optimization](https://mlsanity.com/paper/2310.02524/) (2023)
- [Learning from Few Samples: Transformation-Invariant SVMs with Composition and Locality at Multiple Scales](https://mlsanity.com/paper/2109.12784/) (2021)
- [Measuring dissimilarity with diffeomorphism invariance](https://mlsanity.com/paper/2202.05614/) (2022)
- [Approximating invariant functions with the sorting trick is theoretically justified](https://mlsanity.com/paper/2403.01671/) (2024)

## Abstract (arXiv)

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.