Frame theory and Convolutional
Neural Networks (CNNs)

CNNs perform penalized harmonic
analysis thanks to convolutional (learned) and neural (usually fixed a priori) operators.
Convolutional kernels learned by CNNs
can be classified in 3 main categories: **Meanlets**
(penalized mean operators, behaving as weighted means, smoothing
kernels or scaling functions), **Differencelets**
(penalized differencing operators sharing similar properties as
wavelets and edge detectors) and **Distortlets**
(penalized
distortion operators: neither meanlets, nor differencelets), see reference [1] for details.

*[1] **Frames Learned by Prime Convolution Layers in a Deep Learning Framework**, IEEE TNNLS, 2020*

Rference [1] provides some harmonic content analysis, frame
operator properties and intra-layer correlation structures of convolutional kernels
(that are discovered during training of ImageNet) by CNNs such as ALEXNET,
GOOGLENET, RESNET101 and VGG19.

**Example
of a Table showing the distribution (in percentage) of harmonic content for
layer #1 convolution kernels in R-G-B.**

**Example
of intra-layer inter-kernel correlation structures in R-G-B**

ALEXNET and GOOGLENET show less
correlation between layer #1 convolution kernels than RESNET101 and VGG19. This
implies more ‘redundancy’ for RESNET101 and VGG19 when passing layer #1.

**Conclusion**

The distribution
of harmonic content, as well as other statistics provided by [1] can help
understanding CNN properties such as generalizability and abstraction capabilities.