Strided Maximum Pooling with Overlap

In convolutional neural networks, pooling operations serve as critical downsampling mechanisms that reduce spatial dimensionality while preserving the most salient features. Among pooling variants, strided maximum pooling with overlapping regions offers a unique balance between feature preservation and spatial reduction.

Unlike standard max pooling where adjacent windows are non-overlapping (stride equals kernel size), overlapping pooling uses a stride smaller than the kernel size, causing consecutive pooling windows to share common spatial regions. This overlap creates redundancy in the extracted features, which has been shown to improve generalization and reduce overfitting in deep networks.

Mathematical Formulation:

Given a 4D input tensor X with shape (N, C, H, W) representing:

• N: Batch size (number of samples)
• C: Number of channels (feature maps)
• H: Height of the spatial dimension
• W: Width of the spatial dimension

The overlapping max pooling operation applies a k × k window with stride s (where s < k) across the height and width dimensions. For each window position, the maximum value within that window is selected.

Output Dimension Calculation (Ceiling Mode):

The output spatial dimensions are computed using ceiling mode, which allows partial windows at boundaries:

$$H_{out} = \lceil \frac{H - k}{s} \rceil + 1$$

$$W_{out} = \lceil \frac{W - k}{s} \rceil + 1$$

Key Properties of Overlapping Pooling:

1. Enhanced Translation Invariance: Overlapping regions capture features across multiple spatial positions
2. Richer Feature Maps: Adjacent output positions share input regions, preserving more spatial information
3. Boundary Handling: Ceiling mode ensures all input elements participate in at least one pooling window

Your Task:

Implement a function that performs 2D maximum pooling with overlapping regions on a 4D input tensor. The function should:

• Accept a 4D numpy array, kernel size, and stride as inputs
• Apply max pooling with the specified kernel and stride
• Use ceiling mode for computing output dimensions
• Handle partial windows at boundaries correctly
• Return the pooled 4D output tensor