Good article for buliding an intuition on how to derive the backpropagation formula:https://medium.com/@pavisj/convolutions-and-backpropagations-46026a8f5d2c

University of California, Berkeley

A “filter”, sometimes called a “kernel”, is passed over the image, viewing a few pixels at a time (for example, 3X3 or 5X5). The convolution operation is a dot product of the original pixel values with weights defined in the filter. The results are summed up into one number that represents all the pixels the filter observed.

Convolution Filter (Kernel)

The back-propagation in CNN is done on the same principle of the chain rule, but the the the way it is actually done is not that easy. Of course if we have the gradient of the output we could do bunch of nested loops and calculate the gradients of the parameters and inputs, but this would be crazy inefficient. We need to search a way to vectorize this. We already implemented the forward propagation, and convolution operation is heavily involved in back propagation. The formulas for the gradients of the input are the following:(Suppose L is the cost function, F is the current filter, X is the current input)


Backpropagation in CNN

The cross-correlation operation involves mapping a filter matrix (or kernel) to every possible position on an input matrix. At each position, the corresponding elements of the input and the kernel are multiplied together, and the products are summed to calculate a single entry in the resulting output matrix. While a strict mathematical convolution requires flipping the two-dimensional kernel both horizontally and vertically prior to this process, deep learning implementations typically omit this mirroring step and compute the cross-correlation directly. Despite this technical distinction, it is standard terminology in deep learning literature to refer to the cross-correlation operation simply as a convolution.

The Cross-Correlation (Convolution) Operation

https://medium.com/@RosieCampbell/demystifying-deep-neural-nets-efb726eae941

Demystifying Deep Neural Nets

Three basic strategies for obtaining convolution kernels without supervised training:
- Initializing them randomly
- Setting each kernel to detect edges at a certain orientation or scale
- Item learning the kernels with an unsupervised criterion

Obtaining Kernels without Supervised Training

A convolutional layer performs a cross-correlation operation between its input and a kernel, and subsequently adds a scalar bias to the result to generate an output. During model training, the convolutional layer learns two key parameters: the kernel (weights) and the scalar bias. Prior to training, these kernel weights are generally initialized with random values, analogous to the initialization process in fully connected layers.

Convolutional Layer

The dimensions of a convolution kernel are defined by its height ($$h$$) and width ($$w$$), commonly referred to as an $$h 	imes w$$ convolution kernel. Accordingly, the operation utilizing this kernel is known as an $$h 	imes w$$ convolution. Furthermore, a convolutional layer that employs an $$h 	imes w$$ convolution kernel is simply denoted as an $$h 	imes w$$ convolutional layer.

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