Pooling & spatial downsampling

This feature map has two detections: a strong one (6) in the top-left window, a weaker one (4) in the bottom-right. Toggle between max and average pooling. Max pooling reports the 6 exactly — it just asks "was this feature detected anywhere in the window?" Average pooling reports 3 — it blends the strong detection with the zeros around it, and the signal gets weaker just from being averaged.

Pooling slides a window across a feature map like convolution does, but instead of a learned dot product it applies a fixed reduction — usually the max or the mean — to every value in the window:

• $\text{maxpool}(r,c)$ — the pooled output value at row $r$, column $c$.
• $\text{feature}(\cdot)$ — the input feature map produced by the convolution layer.
• $s$ — the stride: how many positions the window shifts between neighboring windows.
• $r, c$ — the row and column of the output (pooled) position.
• $i, j$ — the row and column offsets ranging over the pooling window.

1. Shrinks the map by the stride

   A $2\times2$ window with stride $2$ turns this $4\times4$ feature map into a $2\times2$ one — a quarter of the values.

2. But keeps what matters

   Those values are chosen so nothing important gets thrown away: taking the max of each window keeps the strongest response, not an arbitrary one.

Move the window to the bottom-right corner. Max pooling reports $4$ — the one real detection there survives untouched. Average pooling reports $1$ — a quarter of $4$, diluted by three zeros. Now check the other two windows: both poolings agree on $0$, because there's nothing there to lose in the first place. The disagreement only shows up exactly where there's a strong, localized signal — which is exactly why max pooling is the default choice after a convolution layer.

1. Top-left window

   Rows 0-1 and columns 0-1 hold the values $2, 2, 2, 6$.

   • Max pooling takes $\max(2,2,2,6) = 6$ — the exact strength of the strongest detection, unchanged.
   • Average pooling takes $\frac{2+2+2+6}{4} = 3$ — half of the true peak, because it's been averaged against three much smaller values.
2. Bottom-right window

   The values $4, 0, 0, 0$ tell the same story at a smaller scale:

   • Max reports $4$
   • Average reports $\frac{4}{4}=1$