When input size is $n_H * n_w * n_c$. the padding output size would be $floor((n_H - f)/s +1) * floor((n_W - f)/s + 1) * n_C$

$n_H$, $n_W$  and  $n_C$  are the hight , wight and channels of the input layer.



The output size of pooling

https://towardsdatascience.com/classifying-skin-lesions-with-convolutional-neural-networks-fc1302c60d54

Classifying Skin Lesions with Convolutional Neural Networks

- It decreases the computational power required to process the data through dimensionality reduction.
- It is useful for extracting dominant features which are rotational and positional invariant, thus maintaining the process of effective model training.

Advantages of Pooling in Convolutional Deep Learning

Types of Pooling Layer in Convolutional Deep Learning

A pooling layer is configured entirely through hyperparameters rather than learned parameters. The key hyperparameters are the pooling window size (which can be specified as a single integer for a square window or as a tuple for a rectangular window with distinct height and width), the stride (the number of elements the window moves per step), the type of aggregation (max or average), and the amount of padding applied to the input borders. Because pooling computes a fixed, deterministic function—either the maximum or the average—there are no weight parameters to learn, and therefore no parameter initialization is needed. A widely adopted default configuration uses a $$2 	imes 2$$ pooling window, which quarters the spatial resolution of the output (halving both height and width), providing an effective balance between downsampling and information retention.

Pooling Parameters and Hyperparameters in Convolutional Deep Learning

1. Reduce parameters. By reducing the dimension of feature map, the parameters required for subsequent layers are effectively reduced

2. Translation variance. It means that for input, when the pixels in the neighborhood are slightly displaced, the output of the pooling layer is unchanged. This enhances the robustness of the network and has a certain anti disturbance effect (translation invariance)

The error of feature extraction mainly comes from two aspects:

(1) The variance of the estimated value increases due to the limited size of the neighborhood;

(2) The convolution layer parameter error causes the offset of the estimated mean.

Generally speaking, mean pooling can reduce the first error and retain more background information of the image. Max pooling can reduce the second error and retain more texture information.

Role of pooling

The forward propagation of a two-dimensional pooling layer can be implemented programmatically without using a kernel. A function `pool2d(X, pool_size, mode)` iterates over the input tensor `X` with a sliding window defined by `pool_size`. At each location, it extracts the sub-tensor and computes either its maximum or mean, depending on the `mode` parameter. Unlike cross-correlation, this computation relies purely on deterministic aggregation over the input regions rather than learnable parameters.

Two-Dimensional Pooling Layer Code Implementation

Pooling and convolutional layers handle multi-channel input data in fundamentally different ways. In a convolutional layer, a multi-channel cross-correlation is performed: the kernel computes a two-dimensional cross-correlation with each input channel separately, but the results from all $$c_	extrm{i}$$ channels are then summed together to produce a single two-dimensional output per output channel. In contrast, the pooling layer processes each input channel independently—it applies the pooling window to every channel in isolation without combining information across channels. As a direct consequence, the number of output channels of a pooling layer always equals the number of input channels, whereas a convolutional layer's output channel count is determined by the number of kernels ($$c_	extrm{o}$$), which is independent of $$c_	extrm{i}$$. This architectural difference means pooling preserves the per-channel feature representations established by preceding convolutional layers, while convolution mixes cross-channel information to create new feature combinations.

Pooling vs. Convolution in Multi-Channel Processing

Pooling operators use a fixed-shape window that slides over all regions of the input tensor based on a specified stride. For each location traversed, this pooling window computes a single output. Unlike convolutional layers that compute cross-correlations using learnable parameters (kernels), the pooling layer contains no parameters. Instead, pooling operations are deterministic, typically calculating the maximum or average value within the pooling window.

University of Michigan - Ann Arbor

Claude

CNNs are made up of three-layer types—convolutional, pooling and fully-connected (FC).

In the convolutional layers, the input matrix is passed through a set of filters that output a feature map. This output is then sent to a pooling layer, which reduces the size of the feature map. This helps reduce the processing time by condensing the map to its most essential information.

The convolutional and pooling processes are repeated several times, with the number of repeats depending on the network, after which the condensed feature map outputs are sent to a series of FC layers. These FC layers then flatten the maps together and compare the probabilities of each feature occurring in conjunction with the others, until the best classification is determined.

Convolutional Neural Networks Architecture

A helpful website for understanding how CNN works:
https://towardsdatascience.com/a-comprehensive-guide-to-convolutional-neural-networks-the-eli5-way-3bd2b1164a53

CNN Reference

A helpful website that introduces neural networks:
https://missinglink.ai/guides/neural-network-concepts/

Neural Network Reference

Dive into Deep Learning

*Pros:*
- The architecture allows CNNs to learn the position and scale of features in a variety of images, making them especially good at the classification of hierarchical or spatial data and the extraction of unlabeled features.
- Because of **Parameter sharing** and **Sparsity of connections** the neural network has significantly fewer parameters that allows it to train a smaller training sets and be less prone to overfitting.

*Cons:*
- Unfortunately, this structure requires CNNs to only accept fixed-size inputs—and it only allows them to provide fixed-size outputs.

Pros and Cons of CNN Architecture

Fully connected layer is a traditional multilayer perceptron structure. Its input is a one-dimensional vector representing the output of the previous layers. Its output is a list of probabilities for different possible labels attached to the image (e.g. dog, cat, bird). The label that receives the highest probability is the classification decision.

Fully Connected Layer - Classification

Here is a clear 3D representation of the convolution Neural Network for detecting a written number. You can see clearly which layer outputs depend on which inputs:
https://www.cs.ryerson.ca/~aharley/vis/conv/

3D Visualization of a Convolution Neural Network

Three classic networks

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)

Pooling Layer in Convolutional Deep Learning

Example of a Convolutional Neural Network Architecture

- Artificial neural network that builds on constructs from pyramid cells in the cerebral cortex
- Residual neural network do this by using shortcuts to jump over some layers 


ResNets Convolutional Neural Network

There are three main classic neural network architectures:
- LeNet-5
- AlexNet
- VGG - 16

Classic Convolutional Neural Network Architectures for Object Detection in Images

The Inception network, also known as GoogLeNet, is a deep convolutional neural network architecture that arranges multiple Inception modules (multi-branch convolutional blocks) into a sequential pipeline for multi-scale feature detection. GoogLeNet uses a stack of $$9$$ Inception blocks organized into three groups with max-pooling between them, and employs global average pooling in its output head to generate predictions. The stem of the network resembles earlier architectures like AlexNet and LeNet. Max-pooling between Inception block groups reduces the spatial dimensionality. The model is computationally complex and involves a large number of relatively arbitrary hyperparameters governing channel counts, the number of blocks before dimensionality reduction, and the relative partitioning of capacity across channels.

Inception Network (GoogLeNet)

The architecture means the overall structure of the network, where most neural networks are organized into groups of units called layers. Different network architecture can have different amount of layers and connections. The main thing to keep in mind when choosing an architecture design, is choosing the depth of the network and the width of each layer.

Architecture Design

We can compare convolution to a fully connected net, but with an inﬁnitely strong prior over its weights: 
                                                                                                                                                                           
1. the weights for one hidden unit must be identical to the weights of its neighbor but shifted in space. 
2. the weights must be zero, except for the small, spatially contiguous receptive ﬁeld assigned to that hidden unit.
                                                                                                                                                                           
Pooling is similar:  
                                                         
1. contains only local interactions and is equivariant to translation
2. each unit should be invariant to small translations.


Convolution and Pooling as an Inﬁnitely Strong Prior

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

Although convolutional neural networks typically contain fewer learnable parameters than comparably deep multi-layer perceptrons, they can still be more computationally expensive to train. The reason is parameter sharing: each convolutional kernel weight participates in many more multiplications because it is applied across every spatial position in the input feature map. As a result, the total number of floating-point operations can exceed that of a similarly structured MLP, even though the CNN stores far fewer unique weights. This makes GPU acceleration especially valuable when training CNNs, since the parallelism of GPUs is well-suited to the large number of repeated multiply–accumulate operations that convolutions require.

CNN Computational Cost versus Parameter Count

In modern deep learning models, particularly Convolutional Neural Networks (CNNs), feature representations are learned automatically rather than being manually engineered. These features are composed hierarchically across multiple jointly learned layers. The lowest layers of the network typically learn to extract basic visual elements, such as edges, colors, and simple textures. Intermediate layers build upon these foundational features to detect more complex structures like object parts, while the highest layers assemble them to recognize entire objects. The final hidden state provides a compact representation that summarizes the image contents, allowing data belonging to different categories to be easily separated by a classifier.

Hierarchical Feature Learning in Vision Networks

The standard building block of early Convolutional Neural Networks consists of a convolutional layer with padding to maintain resolution, a nonlinearity (such as a ReLU), and a pooling layer (such as max-pooling) that reduces the spatial resolution. Because max-pooling typically halves the resolution in each block, this approach imposes a strict limit on network depth. Specifically, a network with an input dimension $$d$$ can accommodate at most $$\log_2 d$$ such convolutional layers before the spatial dimensions are entirely exhausted. For example, on ImageNet data, this architectural design restricts the network to a maximum of $$8$$ convolutional layers.

Spatial Resolution Limit in CNNs

ParNet is a neural network architecture that achieves competitive performance using a much shallower design compared to deep convolutional models like VGG. Instead of relying on a deep, sequential progression of layers, ParNet utilizes a large number of parallel computations to process information. This approach demonstrates a viable alternative to the dominant trend of increasingly deep networks, suggesting that highly parallelized shallow architectures can also be highly effective for modern deep learning tasks.

ParNet Architecture

The Network in Network (NiN) architecture introduces the concept of applying a fully connected layer at each individual pixel location (height and width) of a feature map. Convolutional layers output four-dimensional tensors corresponding to the example, channel, height, and width, whereas traditional fully connected layers expect two-dimensional tensors corresponding to the example and feature. To bridge this gap without flattening the spatial dimensions prematurely, NiN utilizes $$1 	imes 1$$ convolutions. A $$1 	imes 1$$ convolution effectively acts as a fully connected layer operating independently on each pixel's channels across the spatial grid.

Network in Network (NiN) Architecture

ShiftNet is a convolutional neural network architecture designed to increase functional complexity without adding computational cost. It achieves this by mimicking the effects of a standard $$3 \times 3$$ spatial convolution through a simpler mechanism: it explicitly adds spatially shifted activations across different channels. This "shift" operation requires zero floating-point operations and zero parameters, serving as a highly efficient alternative to traditional convolutions.

ShiftNet Architecture

Traditionally, the design of deep neural networks has been a highly manual process, relying heavily on human intuition to find optimal hyperparameters and structural configurations. Because this manual approach is costly in terms of human time and does not guarantee optimal outcomes, researchers developed the notion of network design spaces. Instead of manually designing a single specific network instance, this automated strategy parameterizes a population of possible architectures—a design space—and searches within it to systematically discover high-quality models, such as the RegNetX and RegNetY families.

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