In this paper, we investigate properties and limitations of invariance learned by neural networks from the data compared to the genuine invariance achieved through invariant weight-tying. To do so, we adopt a group theoretical perspective and analyze invariance learning in neural networks without weight-tying constraints. We demonstrate that even when a network learns to correctly classify samples on a group orbit, the underlying decision-making in such a model does not attain genuine invariance. Instead, learned invariance is strongly conditioned on the input data, rendering it unreliable if the input distribution shifts. We next demonstrate how to guide invariance learning toward genuine invariance by regularizing the invariance of a model at the training. To this end, we propose several metrics to quantify learned invariance: (i) predictive distribution invariance, (ii) logit invariance, and (iii) saliency invariance similarity. We show that the invariance learned with the invariance error regularization closely reassembles the genuine invariance of weight-tying models and reliably holds even under a severe input distribution shift. Closer analysis of the learned invariance also reveals the spectral decay phenomenon, when a network chooses to achieve the invariance to a specific transformation group by reducing the sensitivity to any input perturbation.

通过分析神经网络中的不变性学习，我们研究了从数据中学习的不变性与通过不变性权重绑定实现的真实不变性之间的属性和限制。通过引入不变性错误正则化，我们展示了如何指导不变性学习朝着真实不变性的方向发展，并且显示这种学习的不变性在严重的输入分布偏移下也是可靠的。

关于真实的不变性学习的研究，不涉及权重绑定