Convolutional neural network

In, a convolutional neural network (CNN, or ConvNet) is a class of s, most commonly applied to analyzing visual imagery.

CNNs are versions of s. Multilayer perceptrons usually mean fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer. The "fully-connectedness" of these networks makes them prone to data. Typical ways of regularization include adding some form of magnitude measurement of weights to the loss function. However, CNNs take a different approach towards regularization: they take advantage of the hierarchical pattern in data and assemble more complex patterns using smaller and simpler patterns. Therefore, on the scale of connectedness and complexity, CNNs are on the lower extreme.

They are also known as shift invariant or space invariant artificial neural networks (SIANN), based on their shared-weights architecture and characteristics.

Convolutional networks were by  processes in that the connectivity pattern between  resembles the organization of the animal. Individual s respond to stimuli only in a restricted region of the known as the. The receptive fields of different neurons partially overlap such that they cover the entire visual field.

CNNs use relatively little pre-processing compared to other. This means that the network learns the that in traditional algorithms were. This independence from prior knowledge and human effort in feature design is a major advantage.

They have applications in, s, , , and.

Definition
The name “convolutional neural network” indicates that the network employs a mathematical operation called . Convolution is a specialized kind of linear operation. Convolutional networks are simply neural networks that use convolution in place of general matrix multiplication in at least one of their layers

Design
A convolutional neural network consists of an input and an output layer, as well as multiple. The hidden layers of a CNN typically consist of a series of convolutional layers that convolve with a multiplication or other. The activation function is commonly a, and is subsequently followed by additional convolutions such as pooling layers, fully connected layers and normalization layers, referred to as hidden layers because their inputs and outputs are masked by the activation function and final. The final convolution, in turn, often involves in order to more accurately weight the end product.

Though the layers are colloquially referred to as convolutions, this is only by convention. Mathematically, it is technically a sliding dot product or. This has significance for the indices in the matrix, in that it affects how weight is determined at a specific index point.

Convolutional
When programming a CNN, each convolutional layer within a neural network should have the following attributes:


 * Input is a tensor with shape (number of images) x (image width) x (image height) x.
 * Convolutional kernels whose width and height are hyper-parameters, and whose depth must be equal to that of the image. Convolutional layers convolve the input and pass its result to the next layer. This is similar to the response of a neuron in the visual cortex to a specific stimulus.

Each convolutional neuron processes data only for its. Although can be used to learn features as well as classify data, it is not practical to apply this architecture to images. A very high number of neurons would be necessary, even in a shallow (opposite of deep) architecture, due to the very large input sizes associated with images, where each pixel is a relevant variable. For instance, a fully connected layer for a (small) image of size 100 x 100 has 10,000 weights for each neuron in the second layer. The convolution operation brings a solution to this problem as it reduces the number of free parameters, allowing the network to be deeper with fewer parameters. For instance, regardless of image size, tiling regions of size 5 x 5, each with the same shared weights, requires only 25 learnable parameters. In this way, it resolves the vanishing or exploding gradients problem in training traditional multi-layer neural networks with many layers by using.

Pooling
Convolutional networks may include local or global pooling layers to streamline the underlying computation. Pooling layers reduce the dimensions of the data by combining the outputs of neuron clusters at one layer into a single neuron in the next layer. Local pooling combines small clusters, typically 2 x 2. Global pooling acts on all the neurons of the convolutional layer. In addition, pooling may compute a max or an average. Max pooling uses the maximum value from each of a cluster of neurons at the prior layer. Average pooling uses the average value from each of a cluster of neurons at the prior layer.

Fully connected
Fully connected layers connect every neuron in one layer to every neuron in another layer. It is in principle the same as the traditional neural network (MLP). The flattened matrix goes through a fully connected layer to classify the images.

Receptive field
In neural networks, each neuron receives input from some number of locations in the previous layer. In a fully connected layer, each neuron receives input from every element of the previous layer. In a convolutional layer, neurons receive input from only a restricted subarea of the previous layer. Typically the subarea is of a square shape (e.g., size 5 by 5). The input area of a neuron is called its receptive field. So, in a fully connected layer, the receptive field is the entire previous layer. In a convolutional layer, the receptive area is smaller than the entire previous layer.

Weights
Each neuron in a neural network computes an output value by applying a specific function to the input values coming from the receptive field in the previous layer. The function that is applied to the input values is determined by a vector of weights and a bias (typically real numbers). Learning, in a neural network, progresses by making iterative adjustments to these biases and weights.

The vector of weights and the bias are called filters and represent particular s of the input (e.g., a particular shape). A distinguishing feature of CNNs is that many neurons can share the same filter. This reduces because a single bias and a single vector of weights are used across all receptive fields sharing that filter, as opposed to each receptive field having its own bias and vector weighting.

History
CNN design follows vision processing in.

Receptive fields in the visual cortex
Work by and  in the 1950s and 1960s showed that cat and monkey visual es contain neurons that individually respond to small regions of the. Provided the eyes are not moving, the region of visual space within which visual stimuli affect the firing of a single neuron is known as its . Neighboring cells have similar and overlapping receptive fields. Receptive field size and location varies systematically across the cortex to form a complete map of visual space. The cortex in each hemisphere represents the contralateral.

Their 1968 paper identified two basic visual cell types in the brain:


 * s, whose output is maximized by straight edges having particular orientations within their receptive field
 * s, which have larger s, whose output is insensitive to the exact position of the edges in the field.

Hubel and Wiesel also proposed a cascading model of these two types of cells for use in pattern recognition tasks.

Neocognitron, origin of the CNN architecture
The "" was introduced by in 1980. It was inspired by the above-mentioned work of Hubel and Wiesel. The neocognitron introduced the two basic types of layers in CNNs: convolutional layers, and downsampling layers. A convolutional layer contains units whose receptive fields cover a patch of the previous layer. The weight vector (the set of adaptive parameters) of such a unit is often called a filter. Units can share filters. Downsampling layers contain units whose receptive fields cover patches of previous convolutional layers. Such a unit typically computes the average of the activations of the units in its patch. This downsampling helps to correctly classify objects in visual scenes even when the objects are shifted.

In a variant of the neocognitron called the cresceptron, instead of using Fukushima's spatial averaging, J. Weng et al. introduced a method called max-pooling where a downsampling unit computes the maximum of the activations of the units in its patch. Max-pooling is often used in modern CNNs.

Several supervised and unsupervised learning algorithms have been proposed over the decades to train the weights of a neocognitron. Today, however, the CNN architecture is usually trained through.

The is the first CNN which requires units located at multiple network positions to have shared weights. Neocognitrons were adapted in 1988 to analyze time-varying signals.

Time delay neural networks
The (TDNN) was introduced in 1987 by  et al. and was the first convolutional network, as it achieved shift invariance. It did so by utilizing weight sharing in combination with training. Thus, while also using a pyramidal structure as in the neocognitron, it performed a global optimization of the weights, instead of a local one.

TDNNs are convolutional networks that share weights along the temporal dimension. They allow speech signals to be processed time-invariantly. In 1990 Hampshire and Waibel introduced a variant which performs a two dimensional convolution. Since these TDNNs operated on spectrograms the resulting phoneme recognition system was invariant to both, shifts in time and in frequency. This inspired translation invariance in image processing with CNNs. The tiling of neuron outputs can cover timed stages.

TDNNs now achieve the best performance in far distance speech recognition.

Max pooling
In 1990 Yamaguchi et al. introduced the concept of max pooling. They did so by combining TDNNs with max pooling in order to realize a speaker independent isolated word recognition system. In their system they used several TDNNs per word, one for each. The results of each TDNN over the input signal were combined using max pooling and the outputs of the pooling layers were then passed on to networks performing the actual word classification.

Image recognition with CNNs trained by gradient descent
A system to recognize hand-written numbers involved convolutions in which the kernel coefficients had been laboriously hand designed.

et al. (1989) used back-propagation to learn the convolution kernel coefficients directly from images of hand-written numbers. Learning was thus fully automatic, performed better than manual coefficient design, and was suited to a broader range of image recognition problems and image types.

This approach became a foundation of modern.

LeNet-5
LeNet-5, a pioneering 7-level convolutional network by et al. in 1998, that classifies digits, was applied by several banks to recognize hand-written numbers on checks  digitized in 32x32 pixel images. The ability to process higher resolution images requires larger and more layers of convolutional neural networks, so this technique is constrained by the availability of computing resources.

Shift-invariant neural network
Similarly, a shift invariant neural network was proposed by W. Zhang et al. for image character recognition in 1988. The architecture and training algorithm were modified in 1991 and applied for medical image processing and automatic detection of breast cancer in.

A different convolution-based design was proposed in 1988 for application to decomposition of one-dimensional convolved signals via de-convolution. This design was modified in 1989 to other de-convolution-based designs.

Neural abstraction pyramid
The feed-forward architecture of convolutional neural networks was extended in the neural abstraction pyramid by lateral and feedback connections. The resulting recurrent convolutional network allows for the flexible incorporation of contextual information to iteratively resolve local ambiguities. In contrast to previous models, image-like outputs at the highest resolution were generated.

GPU implementations
Although CNNs were invented in the 1980s, their breakthrough in the 2000s required fast implementations on s or s.

In 2004, it was shown by K. S. Oh and K. Jung that standard neural networks can be greatly accelerated on GPUs. Their implementation was 20 times faster than an equivalent implementation on. In 2005, another paper also emphasised the value of for.

The first GPU-implementation of a CNN was described in 2006 by K. Chellapilla et al. Their implementation was 4 times faster than an equivalent implementation on CPU. Subsequent work also used GPUs, initially for other types of neural networks (different from CNNs), especially unsupervised neural networks.

In 2010, Dan Ciresan et al. at showed that even deep standard neural networks with many layers can be quickly trained on GPU by supervised learning through the old method known as. Their network outperformed previous machine learning methods on the handwritten digits benchmark. In 2011, they extended this GPU approach to CNNs, achieving an acceleration factor of 60, with impressive results. In 2011, they used such CNNs on GPU to win an image recognition contest where they achieved superhuman performance for the first time. Between May 15, 2011 and September 30, 2012, their CNNs won no less than four image competitions. In 2012, they also significantly improved on the best performance in the literature for multiple image s, including the, the NORB database, the HWDB1.0 dataset (Chinese characters) and the (dataset of 60000 32x32 labeled ).

Subsequently, a similar GPU-based CNN by Alex Krizhevsky et al. won the 2012. A very deep CNN with over 100 layers by Microsoft won the ImageNet 2015 contest.

Intel Xeon Phi implementations
Compared to the training of CNNs using GPUs, not much attention was given to the Intel Xeon Phi coprocessor. A notable development is a parallelization method for training convolutional neural networks on the Intel Xeon Phi, named Controlled Hogwild with Arbitrary Order of Synchronization (CHAOS). CHAOS exploits both the thread- and SIMD-level parallelism that is available on the Intel Xeon Phi.

Distinguishing features
In the past, traditional (MLP) models have been used for image recognition. However, due to the full connectivity between nodes, they suffered from the, and did not scale well with higher resolution images. A 1000×1000-pixel image with channels has 3 million weights, which is too high to feasibly process efficiently at scale with full connectivity.

For example, in, images are only of size 32×32×3 (32 wide, 32 high, 3 color channels), so a single fully connected neuron in a first hidden layer of a regular neural network would have 32*32*3 = 3,072 weights. A 200×200 image, however, would lead to neurons that have 200*200*3 = 120,000 weights.

Also, such network architecture does not take into account the spatial structure of data, treating input pixels which are far apart in the same way as pixels that are close together. This ignores in image data, both computationally and semantically. Thus, full connectivity of neurons is wasteful for purposes such as image recognition that are dominated by input patterns.

Convolutional neural networks are biologically inspired variants of multilayer perceptrons that are designed to emulate the behavior of a. These models mitigate the challenges posed by the MLP architecture by exploiting the strong spatially local correlation present in natural images. As opposed to MLPs, CNNs have the following distinguishing features:
 * 3D volumes of neurons. The layers of a CNN have neurons arranged in : width, height and depth. The neurons inside a layer are connected to only a small region of the layer before it, called a receptive field. Distinct types of layers, both locally and completely connected, are stacked to form a CNN architecture.
 * Local connectivity: following the concept of receptive fields, CNNs exploit spatial locality by enforcing a local connectivity pattern between neurons of adjacent layers. The architecture thus ensures that the learned "" produce the strongest response to a spatially local input pattern. Stacking many such layers leads to that become increasingly global (i.e. responsive to a larger region of pixel space) so that the network first creates representations of small parts of the input, then from them assembles representations of larger areas.
 * Shared weights: In CNNs, each filter is replicated across the entire visual field. These replicated units share the same parameterization (weight vector and bias) and form a feature map. This means that all the neurons in a given convolutional layer respond to the same feature within their specific response field. Replicating units in this way allows for features to be detected regardless of their position in the visual field, thus constituting a property of.

Together, these properties allow CNNs to achieve better generalization on. Weight sharing dramatically reduces the number of s learned, thus lowering the memory requirements for running the network and allowing the training of larger, more powerful networks.

Building blocks
A CNN architecture is formed by a stack of distinct layers that transform the input volume into an output volume (e.g. holding the class scores) through a differentiable function. A few distinct types of layers are commonly used. These are further discussed below.

Convolutional layer
The convolutional layer is the core building block of a CNN. The layer's parameters consist of a set of learnable (or ), which have a small receptive field, but extend through the full depth of the input volume. During the forward pass, each filter is across the width and height of the input volume, computing the  between the entries of the filter and the input and producing a 2-dimensional  of that filter. As a result, the network learns filters that activate when it detects some specific type of at some spatial position in the input.

Stacking the activation maps for all filters along the depth dimension forms the full output volume of the convolution layer. Every entry in the output volume can thus also be interpreted as an output of a neuron that looks at a small region in the input and shares parameters with neurons in the same activation map.

Local connectivity
When dealing with high-dimensional inputs such as images, it is impractical to connect neurons to all neurons in the previous volume because such a network architecture does not take the spatial structure of the data into account. Convolutional networks exploit spatially local correlation by enforcing a pattern between neurons of adjacent layers: each neuron is connected to only a small region of the input volume.

The extent of this connectivity is a called the  of the neuron. The connections are (along width and height), but always extend along the entire depth of the input volume. Such an architecture ensures that the learnt filters produce the strongest response to a spatially local input pattern.

Spatial arrangement
Three control the size of the output volume of the convolutional layer: the depth,  and zero-padding.


 * The  depth  of the output volume controls the number of neurons in a layer that connect to the same region of the input volume. These neurons learn to activate for different features in the input. For example, if the first convolutional layer takes the raw image as input, then different neurons along the depth dimension may activate in the presence of various oriented edges, or blobs of color.
 * Stride controls how depth columns around the spatial dimensions (width and height) are allocated. When the stride is 1 then we move the filters one pixel at a time. This leads to heavily receptive fields between the columns, and also to large output volumes. When the stride is 2 then the filters jump 2 pixels at a time as they slide around. Similarly, for any integer $S > 0,$  a stride of S causes the filter to be translated by S units at a time per output. In practice, stride lengths of $S \geq 3$  are rare. The receptive fields overlap less and the resulting output volume has smaller spatial dimensions when stride length is increased.
 * Sometimes it is convenient to pad the input with zeros on the border of the input volume. The size of this padding is a third hyperparameter. Padding provides control of the output volume spatial size. In particular, sometimes it is desirable to exactly preserve the spatial size of the input volume.

The spatial size of the output volume can be computed as a function of the input volume size $$W$$, the kernel field size of the convolutional layer neurons $$K$$, the stride with which they are applied $$S$$, and the amount of zero padding $$P$$ used on the border. The formula for calculating how many neurons "fit" in a given volume is given by

$$\frac{W-K+2P}{S} + 1.$$

If this number is not an, then the strides are incorrect and the neurons cannot be tiled to fit across the input volume in a way. In general, setting zero padding to be $P = (K-1)/2$ when the stride is $$S=1$$ ensures that the input volume and output volume will have the same size spatially. However, it's not always completely necessary to use all of the neurons of the previous layer. For example, a neural network designer may decide to use just a portion of padding.

Parameter sharing
A parameter sharing scheme is used in convolutional layers to control the number of free parameters. It relies on one reasonable assumption: if a patch feature is useful to compute at some spatial position, then it should also be useful to compute at other positions. In other words, denoting a single 2-dimensional slice of depth as a depth slice, we constrain the neurons in each depth slice to use the same weights and bias.

Since all neurons in a single depth slice share the same parameters, the forward pass in each depth slice of the convolutional layer can be computed as a of the neuron's weights with the input volume. Therefore, it is common to refer to the sets of weights as a filter (or a ), which is convolved with the input. The result of this convolution is an, and the set of activation maps for each different filter are stacked together along the depth dimension to produce the output volume. Parameter sharing contributes to the of the CNN architecture.

Sometimes, the parameter sharing assumption may not make sense. This is especially the case when the input images to a CNN have some specific centered structure; for which we expect completely different features to be learned on different spatial locations. One practical example is when the inputs are faces that have been centered in the image: we might expect different eye-specific or hair-specific features to be learned in different parts of the image. In that case it is common to relax the parameter sharing scheme, and instead simply call the layer a "locally connected layer".

Pooling layer
Another important concept of CNNs is pooling, which is a form of non-linear. There are several non-linear functions to implement pooling among which max pooling is the most common. It the input image into a set of non-overlapping rectangles and, for each such sub-region, outputs the maximum.

Intuitively, the exact location of a feature is less important than its rough location relative to other features. This is the idea behind the use of pooling in convolutional neural networks. The pooling layer serves to progressively reduce the spatial size of the representation, to reduce the number of parameters, and amount of computation in the network, and hence to also control. It is common to periodically insert a pooling layer between successive convolutional layers in a CNN architecture. The pooling operation provides another form of translation invariance.

The pooling layer operates independently on every depth slice of the input and resizes it spatially. The most common form is a pooling layer with filters of size 2×2 applied with a stride of 2 downsamples at every depth slice in the input by 2 along both width and height, discarding 75% of the activations. In this case, every is over 4 numbers. The depth dimension remains unchanged.

In addition to max pooling, pooling units can use other functions, such as pooling or  pooling. Average pooling was often used historically but has recently fallen out of favor compared to max pooling, which performs better in practice.

Due to the aggressive reduction in the size of the representation, there is a recent trend towards using smaller filters or discarding pooling layers altogether.

"" pooling (also known as RoI pooling) is a variant of max pooling, in which output size is fixed and input rectangle is a parameter.

Pooling is an important component of convolutional neural networks for based on Fast R-CNN architecture.

ReLU layer
ReLU is the abbreviation of, which applies the non-saturating. It effectively removes negative values from an activation map by setting them to zero. It increases the of the  and of the overall network without affecting the receptive fields of the convolution layer.

Other functions are also used to increase nonlinearity, for example the saturating, , and the. ReLU is often preferred to other functions because it trains the neural network several times faster without a significant penalty to accuracy.

Fully connected layer
Finally, after several convolutional and max pooling layers, the high-level reasoning in the neural network is done via fully connected layers. Neurons in a fully connected layer have connections to all activations in the previous layer, as seen in regular (non-convolutional) s. Their activations can thus be computed as an, with followed by a bias offset ( of a learned or fixed bias term).

Loss layer
The "loss layer" specifies how penalizes the deviation between the predicted (output) and  labels and is normally the final layer of a neural network. Various s appropriate for different tasks may be used.

loss is used for predicting a single class of K mutually exclusive classes. loss is used for predicting K independent probability values in $$[0,1]$$. loss is used for to  labels $$(-\infty,\infty)$$.

Choosing hyperparameters
CNNs use more than a standard multilayer perceptron (MLP). While the usual rules for s and constants still apply, the following should be kept in mind when optimizing.

Number of filters
Since feature map size decreases with depth, layers near the input layer will tend to have fewer filters while higher layers can have more. To equalize computation at each layer, the product of feature values va with pixel position is kept roughly constant across layers. Preserving more information about the input would require keeping the total number of activations (number of feature maps times number of pixel positions) non-decreasing from one layer to the next.

The number of feature maps directly controls the capacity and depends on the number of available examples and task complexity.

Filter shape
Common filter shapes found in the literature vary greatly, and are usually chosen based on the dataset.

The challenge is, thus, to find the right level of granularity so as to create abstractions at the proper scale, given a particular dataset, and without.

Max pooling shape
Typical values are 2×2. Very large input volumes may warrant 4×4 pooling in the lower layers. However, choosing larger shapes will dramatically of the signal, and may result in excess. Often, non-overlapping pooling windows perform best.

Regularization methods
is a process of introducing additional information to solve an or to prevent. CNNs use various types of regularization.

Dropout
Because a fully connected layer occupies most of the parameters, it is prone to. One method to reduce overfitting is. At each training stage, individual nodes are either "dropped out" of the net with probability $$1-p$$ or kept with probability $$p$$, so that a reduced network is left; incoming and outgoing edges to a dropped-out node are also removed. Only the reduced network is trained on the data in that stage. The removed nodes are then reinserted into the network with their original weights.

In the training stages, the probability that a hidden node will be dropped is usually 0.5; for input nodes, this should be much lower, intuitively because information is directly lost when input nodes are ignored.

At testing time after training has finished, we would ideally like to find a sample average of all possible $$2^n$$ dropped-out networks; unfortunately this is unfeasible for large values of $$n$$. However, we can find an approximation by using the full network with each node's output weighted by a factor of $$p$$, so the of the output of any node is the same as in the training stages. This is the biggest contribution of the dropout method: although it effectively generates $$2^n$$ neural nets, and as such allows for model combination, at test time only a single network needs to be tested.

By avoiding training all nodes on all training data, dropout decreases overfitting. The method also significantly improves training speed. This makes the model combination practical, even for s. The technique seems to reduce node interactions, leading them to learn more robust features that better generalize to new data.

DropConnect
DropConnect is the generalization of dropout in which each connection, rather than each output unit, can be dropped with probability $$1-p$$. Each unit thus receives input from a random subset of units in the previous layer.

DropConnect is similar to dropout as it introduces dynamic sparsity within the model, but differs in that the sparsity is on the weights, rather than the output vectors of a layer. In other words, the fully connected layer with DropConnect becomes a sparsely connected layer in which the connections are chosen at random during the training stage.

Stochastic pooling
A major drawback to Dropout is that it does not have the same benefits for convolutional layers, where the neurons are not fully connected.

In stochastic pooling, the conventional pooling operations are replaced with a stochastic procedure, where the activation within each pooling region is picked randomly according to a, given by the activities within the pooling region. This approach is free of hyperparameters and can be combined with other regularization approaches, such as dropout and data augmentation.

An alternate view of stochastic pooling is that it is equivalent to standard max pooling but with many copies of an input image, each having small local. This is similar to explicit s of the input images, which delivers excellent performance on the. Using stochastic pooling in a multilayer model gives an exponential number of deformations since the selections in higher layers are independent of those below.

Artificial data
Since the degree of model overfitting is determined by both its power and the amount of training it receives, providing a convolutional network with more training examples can reduce overfitting. Since these networks are usually trained with all available data, one approach is to either generate new data from scratch (if possible) or perturb existing data to create new ones. For example, input images could be asymmetrically cropped by a few percent to create new examples with the same label as the original.

Early stopping
One of the simplest methods to prevent overfitting of a network is to simply stop the training before overfitting has had a chance to occur. It comes with the disadvantage that the learning process is halted.

Number of parameters
Another simple way to prevent overfitting is to limit the number of parameters, typically by limiting the number of hidden units in each layer or limiting network depth. For convolutional networks, the filter size also affects the number of parameters. Limiting the number of parameters restricts the predictive power of the network directly, reducing the complexity of the function that it can perform on the data, and thus limits the amount of overfitting. This is equivalent to a "".

Weight decay
A simple form of added regularizer is weight decay, which simply adds an additional error, proportional to the sum of weights or squared magnitude  of the weight vector, to the error at each node. The level of acceptable model complexity can be reduced by increasing the proportionality constant, thus increasing the penalty for large weight vectors.

L2 regularization is the most common form of regularization. It can be implemented by penalizing the squared magnitude of all parameters directly in the objective. The L2 regularization has the intuitive interpretation of heavily penalizing peaky weight vectors and preferring diffuse weight vectors. Due to multiplicative interactions between weights and inputs this has the useful property of encouraging the network to use all of its inputs a little rather than some of its inputs a lot.

L1 regularization is another common form. It is possible to combine L1 with L2 regularization (this is called ). The L1 regularization leads the weight vectors to become sparse during optimization. In other words, neurons with L1 regularization end up using only a sparse subset of their most important inputs and become nearly invariant to the noisy inputs.

Max norm constraints
Another form of regularization is to enforce an absolute upper bound on the magnitude of the weight vector for every neuron and use to enforce the constraint. In practice, this corresponds to performing the parameter update as normal, and then enforcing the constraint by clamping the weight vector $$\vec{w}$$ of every neuron to satisfy $$\|\vec{w}\|_{2}<c$$. Typical values of $$c$$ are order of 3–4. Some papers report improvements when using this form of regularization.

Hierarchical coordinate frames
Pooling loses the precise spatial relationships between high-level parts (such as nose and mouth in a face image). These relationships are needed for identity recognition. Overlapping the pools so that each feature occurs in multiple pools, helps retain the information. Translation alone cannot extrapolate the understanding of geometric relationships to a radically new viewpoint, such as a different orientation or scale. On the other hand, people are very good at extrapolating; after seeing a new shape once they can recognize it from a different viewpoint.

Currently, the common way to deal with this problem is to train the network on transformed data in different orientations, scales, lighting, etc. so that the network can cope with these variations. This is computationally intensive for large data-sets. The alternative is to use a hierarchy of coordinate frames and to use a group of neurons to represent a conjunction of the shape of the feature and its pose relative to the. The pose relative to retina is the relationship between the coordinate frame of the retina and the intrinsic features' coordinate frame.

Thus, one way of representing something is to embed the coordinate frame within it. Once this is done, large features can be recognized by using the consistency of the poses of their parts (e.g. nose and mouth poses make a consistent prediction of the pose of the whole face). Using this approach ensures that the higher level entity (e.g. face) is present when the lower level (e.g. nose and mouth) agree on its prediction of the pose. The vectors of neuronal activity that represent pose ("pose vectors") allow spatial transformations modeled as linear operations that make it easier for the network to learn the hierarchy of visual entities and generalize across viewpoints. This is similar to the way the human imposes coordinate frames in order to represent shapes.

Image recognition
CNNs are often used in systems. In 2012 an of 0.23 percent on the  was reported. Another paper on using CNN for image classification reported that the learning process was "surprisingly fast"; in the same paper, the best published results as of 2011 were achieved in the MNIST database and the NORB database. Subsequently, a similar CNN called won the 2012.

When applied to, CNNs achieved a large decrease in error rate. Another paper reported a 97.6 percent recognition rate on "5,600 still images of more than 10 subjects". CNNs were used to assess in an objective way after manual training; the resulting system had a very low.

The is a benchmark in object classification and detection, with millions of images and hundreds of object classes. In the ILSVRC 2014, a large-scale visual recognition challenge, almost every highly ranked team used CNN as their basic framework. The winner (the foundation of ) increased the mean average  of object detection to 0.439329, and reduced classification error to 0.06656, the best result to date. Its network applied more than 30 layers. That performance of convolutional neural networks on the ImageNet tests was close to that of humans. The best algorithms still struggle with objects that are small or thin, such as a small ant on a stem of a flower or a person holding a quill in their hand. They also have trouble with images that have been distorted with filters, an increasingly common phenomenon with modern digital cameras. By contrast, those kinds of images rarely trouble humans. Humans, however, tend to have trouble with other issues. For example, they are not good at classifying objects into fine-grained categories such as the particular breed of dog or species of bird, whereas convolutional neural networks handle this.

In 2015 a many-layered CNN demonstrated the ability to spot faces from a wide range of angles, including upside down, even when partially occluded, with competitive performance. The network was trained on a database of 200,000 images that included faces at various angles and orientations and a further 20 million images without faces. They used batches of 128 images over 50,000 iterations.

Video analysis
Compared to image data domains, there is relatively little work on applying CNNs to video classification. Video is more complex than images since it has another (temporal) dimension. However, some extensions of CNNs into the video domain have been explored. One approach is to treat space and time as equivalent dimensions of the input and perform convolutions in both time and space. Another way is to fuse the features of two convolutional neural networks, one for the spatial and one for the temporal stream. (LSTM) units are typically incorporated after the CNN to account for inter-frame or inter-clip dependencies. schemes for training spatio-temporal features have been introduced, based on Convolutional Gated Restricted and Independent Subspace Analysis.

Natural language processing
CNNs have also been explored for. CNN models are effective for various NLP problems and achieved excellent results in, search query retrieval, sentence modeling, classification, prediction and other traditional NLP tasks.

Drug discovery
CNNs have been used in. Predicting the interaction between molecules and biological can identify potential treatments. In 2015, Atomwise introduced AtomNet, the first deep learning neural network for structure-based. The system trains directly on 3-dimensional representations of chemical interactions. Similar to how image recognition networks learn to compose smaller, spatially proximate features into larger, complex structures, AtomNet discovers chemical features, such as, and ing. Subsequently, AtomNet was used to predict novel candidate s for multiple disease targets, most notably treatments for the and.

Health risk assessment and biomarkers of aging discovery
CNNs can be naturally tailored to analyze a sufficiently large collection of data representing one-week-long human physical activity streams augmented by the rich clinical data (including the death register, as provided by, e.g., the  study). A simple CNN was combined with Cox-Gompertz and used to produce a proof-of-concept example of digital  in the form of all-causes-mortality predictor.

Checkers game
CNNs have been used in the game of. From 1999 to 2001, and Chellapilla published papers showing how a convolutional neural network could learn to play checkers using co-evolution. The learning process did not use prior human professional games, but rather focused on a minimal set of information contained in the checkerboard: the location and type of pieces, and the difference in number of pieces between the two sides. Ultimately, the program was tested on 165 games against players and ranked in the highest 0.4%. It also earned a win against the program at its "expert" level of play.

Go
CNNs have been used in. In December 2014, Clark and Storkey published a paper showing that a CNN trained by supervised learning from a database of human professional games could outperform and win some games against  Fuego 1.1 in a fraction of the time it took Fuego to play. Later it was announced that a large 12-layer convolutional neural network had correctly predicted the professional move in 55% of positions, equalling the accuracy of a human player. When the trained convolutional network was used directly to play games of Go, without any search, it beat the traditional search program in 97% of games, and matched the performance of the  program Fuego simulating ten thousand playouts (about a million positions) per move.

A couple of CNNs for choosing moves to try ("policy network") and evaluating positions ("value network") driving MCTS were used by, the first to beat the best human player at the time.

Time Series Forecasting
Recurrent neural networks are generally considered the best neural network architectures for time series forecasting (and sequence modeling in general), but recent studies show that convolutional networks can perform comparably or even better. Dilated convolutions might enable one-dimensional convolutional neural networks to effectively learn time series dependences. Convolutions can be implemented more efficiently than RNN-based solutions, and they do not suffer from vanishing (or exploding) gradients. Convolutional networks can provide an improved forecasting performance when there are multiple similar time series to learn from. CNNs can also be applied to further tasks in time series analysis (e.g., time series classification or quantile forecasting).

Fine-tuning
For many applications, little training data is available. Convolutional neural networks usually require a large amount of training data in order to avoid. A common technique is to train the network on a larger data set from a related domain. Once the network parameters have converged an additional training step is performed using the in-domain data to fine-tune the network weights. This allows convolutional networks to be successfully applied to problems with small training sets.

Human interpretable explanations
End-to-end training and prediction are common practice in. However, human interpretable explanations are required for such as a s. With recent advances in,  and , the most critical spatial regions/temporal instants could be visualized to justify the CNN predictions.

Deep Q-networks
A deep Q-network (DQN) is a type of deep learning model that combines a deep CNN with, a form of. Unlike earlier reinforcement learning agents, DQNs can learn directly from high-dimensional sensory inputs.

Preliminary results were presented in 2014, with an accompanying paper in February 2015. The research described an application to gaming. Other deep reinforcement learning models preceded it.

Deep belief networks
Convolutional deep belief networks (CDBN) have structure very similar to convolutional neural networks and are trained similarly to deep belief networks. Therefore, they exploit the 2D structure of images, like CNNs do, and make use of pre-training like s. They provide a generic structure that can be used in many image and signal processing tasks. Benchmark results on standard image datasets like CIFAR have been obtained using CDBNs.

Notable libraries

 * : A library for convolutional neural networks. Created by the Berkeley Vision and Learning Center (BVLC). It supports both CPU and GPU. Developed in, and has and  wrappers.
 * : Deep learning in and  on multi-GPU-enabled . A general-purpose deep learning library for the JVM production stack running on a C++ scientific computing engine. Allows the creation of custom layers. Integrates with Hadoop and Kafka.
 * A toolkit for making real world machine learning and data analysis applications in C++.
 * : A deep learning toolkit written by Microsoft with several unique features enhancing scalability over multiple nodes. It supports full-fledged interfaces for training in C++ and Python and with additional support for model inference in and Java.
 * -licensed Theano-like library with support for CPU, GPU, Google's proprietary (TPU), and mobile devices.
 * The reference deep-learning library for Python with an API largely compatible with the popular library. Allows user to write symbolic mathematical expressions, then automatically generates their derivatives, saving the user from having to code gradients or backpropagation. These symbolic expressions are automatically compiled to  code for a fast,  implementation.
 * A framework with wide support for machine learning algorithms, written in  and . The main author is Ronan Collobert, and it is now used at Facebook AI Research and Twitter.

Notable APIs

 * A high level API written in for  and  convolutional neural networks.