Spatial Mean Downsampling

Spatial Mean Downsampling is a fundamental dimensionality reduction technique widely employed in modern convolutional neural networks and image processing pipelines. This operation systematically reduces the spatial dimensions of a feature map while preserving the most representative information from localized regions.

Given a 2D numerical matrix (often representing an image or feature map) and a specified window size, the operation divides the input into non-overlapping square regions of the given size. For each region, the arithmetic mean of all contained values is computed, producing a single representative value in the output matrix.

Mathematical Formulation:

For an input matrix X of dimensions H × W and a pooling window of size k × k, the output matrix Y has dimensions (H/k) × (W/k). Each output element is computed as:

$$Y_{i,j} = \frac{1}{k^2} \sum_{p=0}^{k-1} \sum_{q=0}^{k-1} X_{(i \cdot k + p), (j \cdot k + q)}$$

Key Properties:

• Dimensionality Reduction: The spatial dimensions are reduced by a factor equal to the pool size in each dimension
• Translation Invariance: Small shifts in input produce similar outputs, making models robust to positional variations
• Noise Reduction: Averaging naturally smooths out high-frequency noise and minor fluctuations
• Feature Abstraction: Converts fine-grained local details into coarser regional summaries

Your Task: Implement a function that performs spatial mean downsampling on a 2D matrix. The function receives the input matrix and the pooling window size, then returns the downsampled result. You may assume that the input dimensions are evenly divisible by the pool size.