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I am trying to understand how batch normalization (BN) works in CNNs. Suppose I have a feature map tensor $T$ of shape $(N, C, H, W)$

where $N$ is the mini-batch size,

$C$ is the number of channels, and

$H,W$ is the spatial dimension of the tensor.

Then it seems there could a few ways of going about this:

Method 1: $T_{n,c,x,y} := \gamma*\frac {T_{c,x,y} - \mu_{x,y}} {\sqrt{\sigma^2_{x,y} + \epsilon}} + \beta$ where $\mu_{x,y} = \frac{1}{NC}\sum_{n, c} T_{n,c,x,y}$ is the mean for all channels $c$ for each batch element $n$ at spatial location $x,y$ over the minibatch, and

$\sigma^2_{x,y} = \frac{1}{NC} \sum_{n, c} (T_{n, c,x,y}-\mu_{c})^2$ is the variance of the minibatch for all channels $c$ at spatial location $x,y$.

Method 2: $T_{n,c,x,y} := \gamma*\frac {T_{c,x,y} - \mu_{c,x,y}} {\sqrt{\sigma^2_{c,x,y} + \epsilon}} + \beta$ where $\mu_{c,x,y} = \frac{1}{N}\sum_{n} T_{n,c,x,y}$ is the mean for a specific channels $c$ for each batch element $n$ at spatial location $x,y$ over the minibatch, and

$\sigma^2_{c,x,y} = \frac{1}{N} \sum_{n} (T_{n, c,x,y}-\mu_{c})^2$ is the variance of the minibatch for a channel $c$ at spatial location $x,y$.

Method 3: For each channel $c$ we compute the mean/variance over the entire spatial values for $x,y$ and apply the formula as

$T_{n, c,x,y} := \gamma*\frac {T_{n, c,x,y} - \mu_{c}} {\sqrt{\sigma^2_{c} + \epsilon}} + \beta$, where now $\mu_c = \frac{1}{NHW} \sum_{n,x,y} T_{n,c,x,y}$ and $\sigma^2{_c} = \frac{1}{NHW} \sum_{n,x,y} (T_{n,c,x,y}-\mu_c)^2 $

In practice which of these methods is used (if any) are correct for?

The original paper on batch normalization , https://arxiv.org/pdf/1502.03167.pdf , states on page 5 section 3.2, last paragraph, left side of the page:

For convolutional layers, we additionally want the normalization to obey the convolutional property – so that different elements of the same feature map, at different locations, are normalized in the same way. To achieve this, we jointly normalize all the activations in a minibatch, over all locations. In Alg. 1, we let $\mathcal{B}$ be the set of all values in a feature map across both the elements of a mini-batch and spatial locations – so for a mini-batch of size $m$ and feature maps of size $p \times q$, we use the effective mini-batch of size $m^\prime = \vert \mathcal{B} \vert = m \cdot pq$. We learn a pair of parameters $\gamma^{(k)}$ and $\beta^{(k)}$ per feature map, rather than per activation. Alg. 2 is modified similarly, so that during inference the BN transform applies the same linear transformation to each activation in a given feature map.

I'm not sure what the authors mean by "per feature map", does this mean per channel?

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  • $\begingroup$ Are you proposing the three methods or are you pulling them out of a document? If you are pulling them from somewhere else please cite them. Also, you want the variance and not the standard deviation in the denominator. $\endgroup$ – Brian O'Donnell Jan 20 at 1:01
  • $\begingroup$ @BrianO'Donnell edited. $\endgroup$ – IntegrateThis Jan 20 at 1:11

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