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In a convolutional neural network, when we apply the convolution on a $5 \times 5$ image with $3 \times 3$ kernel, with stride $1$, we should get only one $4 \times 4$ as output. In most of the CNN tutorials, we are having $4 \times 4 \times m$ as output. I don't know how we are getting a three-dimensional output and I don't know how we need to calculate $m$. How is $m$ determined? Why do we get a three-dimensional output after a convolutional layer?

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If you have a $h_i \times w_i \times d_i$ input, where $h_i, w_i$ and $d_i$ respectively refer to the height, width and depth of the input, then we usually apply $m$ $h_k \times w_k \times d_i$ kernels (or filters) to this input (with the appropriate stride and padding), where $m$ is usually a hyper-parameter. So, after the application of $m$ kernels, you will obtain $m$ $h_o \times w_o \times 1$ so-called feature maps (also known as activation maps), which are usually concatenated along the depth dimension, hence your output will have a depth of $m$ (given that the application of a kernel to the input usually produces a two-dimensional output). For this reason, the output is usually referred to as output volume.

In the context of CNNs, the kernels are learned, so they are not constant (at least, during the learning process, but, after training, they usually remain constant, unless you perform continual lifelong learning). Each kernel will be different from any other kernel, so each kernel will be doing a different convolution with the input (with respect to the other kernels), therefore, each kernel will be responsible for filtering (or detecting) a specific and different (with respect to the other kernels) feature of the input, which can, for example, be the initial image or the output of another convolutional layer.

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Why do we get a three-dimensional output after a convolutional layer?

During a search for an optimal convolution kernel via gradient descent or some other method there must be at least one additional dimension to represent trials. It is most often one. If the input is in $\mathcal{R}^n$ space, then the output of the convolution operation is $\mathcal{R}^{n+1}$ space. However, this is not often (or ever) the output of the learning system using a convolutional layer, since the final layer in the conventional deep network designs used today is not the convolution layer.

In the case in this question, $m$ represents the number of discrete kernels tried, usually in rapid succession in most algorithms and hardware acceleration scenarios. It is using the results in this $\mathcal{R}^{n+1}$ space that the corrective mechanism requires to converge on an optimum efficiently.

By corrective mechanism is meant whatever corrects the assumptions made for the next set of kernels to be tried. This mechanism often involves gradient descent and back propagation in an artificial network design. It is the learning algorithm or, in the case of hardware acceleration, the learning circuit.

The value of $m$ is not arbitrary, but is largely problem dependent and based on hardware and execution environment. If $m$ is too large, then too much convolution work is performed before the correction is made. If $m$ is too small, then whatever efficiency is gained by grouping kernel tries together in time is lost. There may be a formula to find $m$, but it will be based on platform dependent metrics and can be found by trying several $m$ values and determining the $m$ providing the lowest convergence time.

The value of $m$ can also affect accuracy and reliability of convergence. Predicting this effect is not straightforward. Such prediction, which would allow the automated selection of hyper-parameters like $m$, is of interest to many researchers for obvious reasons. It has been and will probably continue to be an objective of AI development to remove the need for human intervention in AI system applications.

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