[论文解读] Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations

Alberto Bietti, Julien Mairal|arXiv (Cornell University)|Jun 9, 2017

Neural Networks and Applications参考文献 56被引用 58

一句话总结

本文开发了一个多层卷积核（CKN）框架，用于研究不变性、对变形的稳定性，以及深度卷积表示的复杂性，并将核的 RKHS 范数与 CNN 的泛化及稳定性联系起来。

ABSTRACT

The success of deep convolutional architectures is often attributed in part to their ability to learn multiscale and invariant representations of natural signals. However, a precise study of these properties and how they affect learning guarantees is still missing. In this paper, we consider deep convolutional representations of signals; we study their invariance to translations and to more general groups of transformations, their stability to the action of diffeomorphisms, and their ability to preserve signal information. This analysis is carried by introducing a multilayer kernel based on convolutional kernel networks and by studying the geometry induced by the kernel mapping. We then characterize the corresponding reproducing kernel Hilbert space (RKHS), showing that it contains a large class of convolutional neural networks with homogeneous activation functions. This analysis allows us to separate data representation from learning, and to provide a canonical measure of model complexity, the RKHS norm, which controls both stability and generalization of any learned model. In addition to models in the constructed RKHS, our stability analysis also applies to convolutional networks with generic activations such as rectified linear units, and we discuss its relationship with recent generalization bounds based on spectral norms.

研究动机与目标

• Characterize invariance and stability properties of deep convolutional representations independent of training data.
• Analyze the functional space (RKHS) induced by a multilayer kernel representation and its relation to CNNs.
• Provide a norm-based measure of model complexity that controls stability and generalization.
• Extend stability analysis to generic CNN activations and relate to existing generalization bounds.
• Discuss how representation regularization via RKHS norms can improve learning outcomes.

提出的方法

• Define a multilayer convolutional kernel representation built from patch extraction, kernel mapping, and pooling operators.
• Use homogeneous dot-product kernels to map patches into RKHS and ensure norm preservation and non-expansiveness.
• Construct a hierarchical kernel with pooling to increase translation invariance and stability to diffeomorphisms.
• Show that the RKHS contains CNNs with smooth homogeneous activations and bound their RKHS norms through layerwise norms of convolutional filters.
• Extend analysis to CNNs with non-smooth activations and relate stability to product of spectral norms.

实验结果

研究问题

• RQ1 How does a multilayer kernel representation provide translation invariance and stability to diffeomorphisms for deep convolutional architectures?
• RQ2 What is the nature of the RKHS induced by the multilayer kernel, and which CNNs reside in it?
• RQ3 How can the RKHS norm serve as a canonical measure of model complexity influencing stability and generalization?
• RQ4 How do kernel approximations (CKNs) affect invariance and stability guarantees?
• RQ5 Can group-equivariant extensions yield invariance to broader transformation groups while maintaining stability?

主要发现

• The multilayer kernel representation preserves signal information and achieves near translation invariance and deformation stability under defined norms.
• The RKHS corresponding to the multilayer kernel contains a large class of CNNs with smooth homogeneous activations, enabling a norm-based complexity control.
• A bound is established on the RKHS norm of a CNN built in the RKHS in terms of layerwise spectral and Frobenius norms and activation-specific complexity factors.
• Stability results extend to kernel approximations used in CKNs, with corresponding trade-offs in information preservation.
• For generic activations like ReLU, similar stability properties can be discussed via the product of spectral norms, linking to existing generalization bounds.