2D Max Pooling Backward Layer

The forward two-dimensional (2D) max pooling layer is a form of non-linear downsampling of an input tensor XR n 1 x n 2 x ... x n p . 2D max pooling partitions the input tensor data into 2D subtensors along dimensions k 1 and k 2, selects an element with the maximal numeric value in each subtensor, and transforms the input tensor to the output tensor Y by replacing each subtensor with its maximum element. For more details, see Forward 2D Max Pooling Layer.

The backward 2D max pooling layer back-propagates the input gradient GR 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 GR l 1 x ... x l p with the gradient computed on the preceding layer

  • Dimensions k 1 and k 2 along which the kernel is applied

  • Kernel sizes m 1 and m2: where p 1 and p 2 are paddings

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 and s 2 are strides

If m 1 > s 1 and/or m 2 > s 2 and if overlapping subtensors are represented with the same maximum located at the same position in the input tensor X, the gradient value z at this position is the sum of input gradients g at the respective positions, as shown in the following figure for m 1 = 3, m 2 = 2, and s 1 = 2:
Backward Two-dimensional Max Pooling Layer

For more complete information about compiler optimizations, see our Optimization Notice.
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