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Normalize data across all observations for each channel independently

The batch normalization operation normalizes the input data
across all observations for each channel independently. To speed up training of the
convolutional neural network and reduce the sensitivity to network initialization, use batch
normalization between convolution and nonlinear operations such as `relu`

.

After normalization, the operation shifts the input by a learnable offset *β* and scales it by a learnable scale factor *γ*.

The `batchnorm`

function applies the batch normalization operation to
`dlarray`

data.
Using `dlarray`

objects makes working with high
dimensional data easier by allowing you to label the dimensions. For example, you can label
which dimensions correspond to spatial, time, channel, and batch dimensions using the
`"S"`

, `"T"`

, `"C"`

, and
`"B"`

labels, respectively. For unspecified and other dimensions, use the
`"U"`

label. For `dlarray`

object functions that operate
over particular dimensions, you can specify the dimension labels by formatting the
`dlarray`

object directly, or by using the `DataFormat`

option.

**Note**

To apply batch normalization within a `layerGraph`

object
or `Layer`

array, use
`batchNormalizationLayer`

.

applies the batch normalization operation to the input data `dlY`

= batchnorm(`dlX`

,`offset`

,`scaleFactor`

)`dlX`

using
the population mean and variance of the input data and the specified offset and scale
factor.

The function normalizes over the `'S'`

(spatial),
`'T'`

(time), `'B'`

(batch), and `'U'`

(unspecified) dimensions of `dlX`

for each channel in the
`'C'`

(channel) dimension, independently.

For unformatted input data, use the `'DataFormat'`

option.

`[`

applies the batch normalization operation and also returns the population mean and variance
of the input data `dlY`

,`popMu`

,`popSigmaSq`

] = batchnorm(`dlX`

,`offset`

,`scaleFactor`

)`dlX`

.

`[`

applies the batch normalization operation and also returns the updated moving mean and
variance statistics. `dlY`

,`updatedMu`

,`updatedSigmaSq`

] = batchnorm(`dlX`

,`offset`

,`scaleFactor`

,`runningMu`

,`runningSigmaSq`

)`runningMu`

and `runningSigmaSq`

are the mean and variance values after the previous training iteration, respectively.

Use this syntax to maintain running values for the mean and variance statistics during training. When you have finished training, use the final updated values of the mean and variance for the batch normalization operation during prediction and classification.

applies the batch normalization operation using the mean `dlY`

= batchnorm(`dlX`

,`offset`

,`scaleFactor`

,`trainedMu`

,`trainedSigmaSq`

)`trainedMu`

and
variance `trainedSigmaSq`

.

Use this syntax during classification and prediction, where
`trainedMu`

and `trainedSigmaSq`

are the final
values of the mean and variance after you have finished training, respectively.

`[___] = batchnorm(___,'DataFormat',FMT)`

applies the batch normalization operation to unformatted input data with format specified by
`FMT`

using any of the input or output combinations in previous syntaxes.
The output `dlY`

is an unformatted `dlarray`

object with
dimensions in the same order as `dlX`

. For example,
`'DataFormat','SSCB'`

specifies data for 2-D image input with the format
`'SSCB'`

(spatial, spatial, channel, batch).

`[___] = batchnorm(___,`

specifies additional options using one or more name-value pair arguments. For example,
`Name,Value`

)`'MeanDecay',0.3`

sets the decay rate of the moving average
computation.

The batch normalization operation normalizes the elements
*x _{i}* of the input by first calculating the mean

$$\widehat{{x}_{i}}=\frac{{x}_{i}-{\mu}_{B}}{\sqrt{{\sigma}_{B}^{2}+\u03f5}},$$

where *ϵ* is a constant that improves numerical
stability when the variance is very small.

To allow for the possibility that inputs with zero mean and unit variance are not optimal for the operations that follow batch normalization, the batch normalization operation further shifts and scales the activations using the transformation

$${y}_{i}=\gamma {\widehat{x}}_{i}+\beta ,$$

where the offset *β* and scale factor
*γ* are learnable parameters that are updated during network
training.

To make predictions with the network after training, batch normalization requires a fixed mean and variance to normalize the data. This fixed mean and variance can be calculated from the training data after training, or approximated during training using running statistic computations.

`relu`

| `fullyconnect`

| `dlconv`

| `dlarray`

| `dlgradient`

| `dlfeval`

| `groupnorm`

| `layernorm`