Contents

# Three-Dimensional Max Pooling Backward Layer

The forward three-dimensional (3D) max pooling layer is a form of non-linear downsampling of an input tensor
X
R
n
1
x
n
2
x ... x
n
p
. 3D max pooling partitions the input tensor data into 3D subtensors along dimensions
k
1
,
k
2
, and
k
3
, selects an element with the maximal numeric value in each subtensor, and transforms the input tensor to the output tensor by replacing each subtensor with its maximum element. For more details, see Forward 3D Max Pooling Layer.
The backward 3D max pooling layer back-propagates the input gradient
G
R
l
1
x ... x
l
p
computed on the preceding layer. The backward layer propagates to the next layer only the elements of the gradient that correspond to the maximum values pooled from subtensors in the forward computation step.

## Problem Statement

Given:
• p
-dimensional tensor
G
R
l
1
x ... x
l
p
with the gradient computed on the preceding layer
• Dimensions
k
1
,
k
2
, and
k
3
along which the kernel is applied
• Kernel sizes
m
1
, m
2
, and
m
3
:
where
p
1
,
p
2
and
p
3
The problem is to compute the value tensor Z = (
z
i
1
...
i
p
) ∈
R
n
1
x ... x
n
p
such that: where:
• s
1
,
s
2
, and
s
3
are strides
If
m
1
>
s
1
,
m
2
>
s
2
, or
m
3
>
s
3
(including the cases where any of these conditions are met simultaneously) and if overlapping subtensors are represented with the same maximum located at the same position in the input tensor
X