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## Specify Layers of Convolutional Neural Network

The first step of creating and training a new convolutional neural network (ConvNet) is to define the network architecture. This topic explains the details of ConvNet layers, and the order they appear in a ConvNet.

The architecture of a ConvNet can vary depending on the types and numbers of layers included. The types and number of layers included depends on the particular application or data. For example, if you have categorical responses, you must have a softmax layer and a classification layer, whereas if your response is continuous, you must have a regression layer at the end of the network. A smaller network with only one or two convolutional layers might be sufficient to learn on a small number of grayscale image data. On the other hand, for more complex data with millions of colored images, you might need a more complicated network with multiple convolutional and fully connected layers.

You can define the layers of a convolutional neural network in MATLAB® in an array format, for example,

```layers = [ imageInputLayer([28 28 1]) convolution2dLayer(3,16,'Padding',1) batchNormalizationLayer reluLayer maxPooling2dLayer(2,'Stride',2) convolution2dLayer(3,32,'Padding',1) batchNormalizationLayer reluLayer fullyConnectedLayer(10) softmaxLayer classificationLayer];```

`layers` is an array of `Layer` objects. `layers` becomes an input for the training function `trainNetwork`.

### Image Input Layer

The image input layer defines the size of the input images of a convolutional neural network and contains the raw pixel values of the images. You can add an input layer using the `imageInputLayer` function. Specify the image size using the `inputSize` argument. The size of an image corresponds to the height, weight, and the number of color channels of that image. For example, for a grayscale image, the number of channels is 1, and for a color image it is 3.

This layer can also perform data normalization by subtracting the mean image of the training set from every input image.

### Convolutional Layer

Filters and Stride: A convolutional layer consists of neurons that connect to subregions of the input images or the outputs of the layer before it. A convolutional layer learns the features localized by these regions while scanning through an image. You can specify the size of these regions using the `filterSize` input argument when you create the layer using the `convolution2dLayer` function.

For each region, the `trainNetwork` function computes a dot product of the weights and the input, and then adds a bias term. A set of weights that are applied to a region in the image is called a filter. The filter moves along the input image vertically and horizontally, repeating the same computation for each region, that is, convolving the input. The step size with which it moves is called a stride. You can specify this step size with the `Stride` name-value pair argument. These local regions that the neurons connect to might overlap depending on the `filterSize` and `'Stride'` values.

The number of weights used for a filter is h*w*c, where h is the height, and w is the width of the filter size, and c is the number of channels in the input (for example, if the input is a color image, the number of color channels is 3). The number of filters determines the number of channels in the output of a convolutional layer. Specify the number of filters using the `numFilters` argument of `convolution2dLayer`.

Feature Maps: As a filter moves along the input, it uses the same set of weights and bias for the convolution, forming a feature map. Hence, the number of feature maps a convolutional layer has is equal to the number of filters (number of channels). Each feature map has a different set of weights and a bias. So, the total number of parameters in a convolutional layer is ((h*w*c + 1)*Number of Filters), where 1 is for the bias.

Zero Padding: You can also apply zero padding to input image borders vertically and horizontally using the `'Padding'` name-value pair argument. Padding is basically adding rows or columns of zeros to the borders of an image input. It helps you control the output size of the layer.

Output Size: The output height and width of a convolutional layer is (Input SizeFilter Size + 2*Padding)/Stride + 1. This value must be an integer for the whole image to be fully covered. If the combination of these parameters does not lead the image to be fully covered, the software by default ignores the remaining part of the image along the right and bottom edge in the convolution.

Number of Neurons: The product of the output height and width gives the total number of neurons in a feature map, say Map Size. The total number of neurons (output size) in a convolutional layer, then, is Map Size*Number of Filters.

For example, suppose that the input image is a 28-by-28-by-3 color image. For a convolutional layer with 16 filters, and a filter size of 8-by-8, the number of weights per filter is 8*8*3 = 192, and the total number of parameters in the layer is (192+1) * 16 = 3088. Assuming stride is 4 in each direction and there is no zero padding, the total number of neurons in each feature map is 6-by-6 ((28 – 8+0)/4 + 1 = 6). Then, the total number of neurons in the layer is 6*6*16 = 256.

Learning Parameters: You can also adjust the learning rates and regularization parameters for this layer using the related name-value pair arguments while defining the convolutional layer. If you choose not to adjust them, `trainNetwork` uses the global training parameters defined by `trainingOptions` function. For details on global and layer training options, see Set Up Parameters and Train Convolutional Neural Network.

A convolutional neural network can consist of one or multiple convolutional layers. The number of convolutional layers depends on the amount and complexity of the data.

### Batch Normalization Layer

Use batch normalization layers between convolutional layers and nonlinearities such as ReLU layers to speed up network training and reduce the sensitivity to network initialization. The layer first normalizes the activations of each channel by subtracting the mini-batch mean and dividing by the mini-batch standard deviation. Then, the layer shifts the input by an offset β and scales it by a scale factor γ. β and γ are themselves learnable parameters that are updated during network training. Create a batch normalization layer using `batchNormalizationLayer`.

Batch normalization layers normalize the activations and gradients propagating through a neural network, making network training an easier optimization problem. To take full advantage of this fact, you can try increasing the learning rate. Since the optimization problem is easier, the parameter updates can be larger and the network can learn faster. You can also try reducing the L2 and dropout regularization. With batch normalization layers, the activations of a specific image are not deterministic, but instead depend on which images happen to appear in the same mini-batch. To take full advantage of this regularizing effect, try shuffling the training data before every training epoch. To specify how often to shuffle the data during training, use the `'Shuffle'` name-value pair argument of `trainingOptions`.

### ReLU Layer

Convolutional and batch normalization layers are usually followed by a nonlinear activation function such as a rectified linear unit (ReLU), specified by a ReLU layer. Create a ReLU layer using the `reluLayer` function. A ReLU layer performs a threshold operation to each element, where any input value less than zero is set to zero, that is,

`$f\left(x\right)=\left\{\begin{array}{cc}x,& x\ge 0\\ 0,& x<0\end{array}.$`
The ReLU layer does not change the size of its input.

There are extensions of the standard ReLU layer that perform slightly different operations and can improve performance for some applications. A leaky ReLU layer multiplies input values less than zero by a fixed scalar, allowing negative inputs to “leak” into the output. Use the `leakyReluLayer` function to create a leaky ReLU layer. A clipped ReLU layer sets negative inputs to zero, but also sets input values above a clipping ceiling equal to that clipping ceiling. This clipping prevents the output from becoming too large. Use the `clippedReluLayer` function to create a clipped ReLU layer.

### Cross Channel Normalization (Local Response Normalization) Layer

This layer performs a channel-wise local response normalization. It usually follows the ReLU activation layer. Create this layer using the `crossChannelNormalizationLayer` function. This layer replaces each element with a normalized value it obtains using the elements from a certain number of neighboring channels (elements in the normalization window). That is, for each element $x$ in the input, `trainNetwork` computes a normalized value ${x}^{\text{'}}$ using

`${x}^{\text{'}}=\frac{x}{{\left(K+\frac{\alpha *ss}{windowChannelSize}\right)}^{\beta }},$`
where K, α, and β are the hyperparameters in the normalization, and ss is the sum of squares of the elements in the normalization window [2]. You must specify the size of the normalization window using the `windowChannelSize` argument of the `crossChannelNormalizationLayer` function. You can also specify the hyperparameters using the `Alpha`, `Beta`, and `K` name-value pair arguments.

The previous normalization formula is slightly different than what is presented in [2]. You can obtain the equivalent formula by multiplying the `alpha` value by the `windowChannelSize`.

### Max- and Average-Pooling Layers

Max- and average-pooling layers follow the convolutional layers for down-sampling, hence, reducing the number of connections to the following layers (usually a fully connected layer). They do not perform any learning themselves, but reduce the number of parameters to be learned in the following layers. They also help reduce overfitting. Create these layers using the `maxPooling2dLayer` and `averagePooling2dLayer` functions.

A max-pooling layer returns the maximum values of rectangular regions of its input. The size of the rectangular regions is determined by the `poolSize` argument of `maxPoolingLayer`. For example, if `poolSize` equals `[2,3]`, then the layer returns the maximum value in regions of height 2 and width 3.

Similarly, the average-pooling layer outputs the average values of rectangular regions of its input. The size of the rectangular regions is determined by the `poolSize` argument of `averagePoolingLayer`. For example, if `poolSize` is [2,3], then the layer returns the average value of regions of height 2 and width 3. The `maxPoolingLayer` and `averagepoolingLayer` functions scan through the input horizontally and vertically in step sizes you can specify using the `'Stride'` name-value pair argument of either function. If the `poolSize` is smaller than or equal to the `Stride`, then the pooling regions do not overlap.

For nonoverlapping regions (`poolSize` and `Stride` are equal), if the input to the pooling layer is n-by-n, and the pooling region size is h-by-h, then the pooling layer down-samples the regions by h [6]. That is, the output of a max- or average-pooling layer for one channel of a convolutional layer is n/h-by-n/h. For overlapping regions, the output of a pooling layer is (Input SizePool Size + 2*Padding)/Stride + 1.

### Dropout Layer

A dropout layer randomly sets the layer’s input elements to zero with a given probability. Create a dropout layer using the `dropoutLayer` function.

Although the output of a dropout layer is equal to its input, this operation corresponds to temporarily dropping a randomly chosen unit and all of its connections from the network during training. So, for each new input element, `trainNetwork` randomly selects a subset of neurons, forming a different layer architecture. These architectures use common weights, but because the learning does not depend on specific neurons and connections, the dropout layer might help prevent overfitting [7], [2]. Similar to max- or average-pooling layers, no learning takes place in this layer.

### Fully Connected Layer

The convolutional (and down-sampling) layers are followed by one or more fully connected layers. Create a fully connected layer using the `fullyConnectedLayer` function.

As the name suggests, all neurons in a fully connected layer connect to all the neurons in the previous layer. This layer combines all of the features (local information) learned by the previous layers across the image to identify the larger patterns. For classification problems, the last fully connected layer combines the features to classify the images. This is the reason that the `outputSize` argument of the last fully connected layer of the network is equal to the number of classes of the data set. For regression problems, the output size must be equal to the number of response variables.

You can also adjust the learning rate and the regularization parameters for this layer using the related name-value pair arguments when creating the fully connected layer. If you choose not to adjust them, then `trainNetwork` uses the global training parameters defined by the `trainingOptions` function. For details on global and layer training options, see Set Up Parameters and Train Convolutional Neural Network.

### Output Layers

#### Softmax and Classification Layers

For classification problems, a softmax layer and then a classification layer must follow the final fully connected layer. You can create these layers using the `softmaxLayer` and `classificationLayer` functions, respectively.

The output unit activation function is the softmax function:

`${y}_{r}\left(x\right)=\frac{\mathrm{exp}\left({a}_{r}\left(x\right)\right)}{\sum _{j=1}^{k}\mathrm{exp}\left({a}_{j}\left(x\right)\right)},$`

where $0\le {y}_{r}\le 1$ and $\sum _{j=1}^{k}{y}_{j}=1$.

The softmax function is the output unit activation function after the last fully connected layer for multi-class classification problems:

`$P\left({c}_{r}|x,\theta \right)=\frac{P\left(x,\theta |{c}_{r}\right)P\left({c}_{r}\right)}{\sum _{j=1}^{k}P\left(x,\theta |{c}_{j}\right)P\left({c}_{j}\right)}=\frac{\mathrm{exp}\left({a}_{r}\left(x,\theta \right)\right)}{\sum _{j=1}^{k}\mathrm{exp}\left({a}_{j}\left(x,\theta \right)\right)},$`
where $0\le P\left({c}_{r}|x,\theta \right)\le 1$ and $\sum _{j=1}^{k}P\left({c}_{j}|x,\theta \right)=1$. Moreover, ${a}_{r}=\mathrm{ln}\left(P\left(x,\theta |{c}_{r}\right)P\left({c}_{r}\right)\right)$, $P\left(x,\theta |{c}_{r}\right)$ is the conditional probability of the sample given class r, and $P\left({c}_{r}\right)$ is the class prior probability.

The softmax function is also known as the normalized exponential and can be considered the multi-class generalization of the logistic sigmoid function [8].

A classification output layer must follow the softmax layer. In the classification output layer, `trainNetwork` takes the values from the softmax function and assigns each input to one of the k mutually exclusive classes using the cross entropy function for a 1-of-k coding scheme [8]:

`$E\left(\theta \right)=-\sum _{i=1}^{n}\sum _{j=1}^{k}{t}_{ij}\mathrm{ln}{y}_{j}\left({x}_{i},\theta \right),$`
where${t}_{ij}$ is the indicator that the ith sample belongs to the jth class, $\theta$ is the parameter vector. ${y}_{j}\left({x}_{i},\theta \right)$is the output for sample i, which in this case, is the value from the softmax function. That is, it is the probability that the network associates the ith input with class j, $P\left({t}_{j}=1|{x}_{i}\right)$.

#### Regression Layer

You can also use ConvNets for regression problems, where the target (output) variable is continuous. In such cases, a regression output layer must follow the final fully connected layer. You can create a regression layer using the `regressionLayer` function. The default loss function for a regression layer is the mean squared error:

`$MSE=E\left(\theta \right)=\sum _{i=1}^{n}\frac{{\left({t}_{i}-{y}_{i}\right)}^{2}}{n},$`

where ${t}_{i}$ is the target output, and ${y}_{i}$ is the network’s prediction for the response variable corresponding to observation i.

## References

[1] Murphy, K. P. Machine Learning: A Probabilistic Perspective. Cambridge, Massachusetts: The MIT Press, 2012.

[2] Krizhevsky, A., I. Sutskever, and G. E. Hinton. "ImageNet Classification with Deep Convolutional Neural Networks. " Advances in Neural Information Processing Systems. Vol 25, 2012.

[3] LeCun, Y., Boser, B., Denker, J.S., Henderson, D., Howard, R.E., Hubbard, W., Jackel, L.D., et al. ''Handwritten Digit Recognition with a Back-propagation Network.'' In Advances of Neural Information Processing Systems, 1990.

[4] LeCun, Y., L. Bottou, Y. Bengio, and P. Haffner. ''Gradient-based Learning Applied to Document Recognition.'' Proceedings of the IEEE. Vol 86, pp. 2278–2324, 1998.

[5] Nair, V. and G. E. Hinton. "Rectified linear units improve restricted boltzmann machines." In Proc. 27th International Conference on Machine Learning, 2010.

[6] Nagi, J., F. Ducatelle, G. A. Di Caro, D. Ciresan, U. Meier, A. Giusti, F. Nagi, J. Schmidhuber, L. M. Gambardella. ''Max-Pooling Convolutional Neural Networks for Vision-based Hand Gesture Recognition''. IEEE International Conference on Signal and Image Processing Applications (ICSIPA2011), 2011.

[7] Srivastava, N., G. Hinton, A. Krizhevsky, I. Sutskever, R. Salakhutdinov. "Dropout: A Simple Way to Prevent Neural Networks from Overfitting." Journal of Machine Learning Research. Vol. 15, pp. 1929-1958, 2014.

[8] Bishop, C. M. Pattern Recognition and Machine Learning. Springer, New York, NY, 2006.

[9] Ioffe, Sergey, and Christian Szegedy. "Batch normalization: Accelerating deep network training by reducing internal covariate shift." preprint, arXiv:1502.03167 (2015).

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