convolution

Convolution, Padding, Stride, and Pooling

Definition

The core spatial operations in CNNs — convolution applies a learnable filter over local receptive fields; padding preserves spatial dimensions; stride controls the step size; pooling downsamples feature maps.

Intuition

Convolution detects local patterns (edges, textures) using shared weights, making the operation translation-equivariant and vastly more parameter-efficient than fully connected layers. Pooling introduces translation invariance and reduces computation. Many deep learning libraries implement cross-correlation (no kernel flip), though the term “convolution” is used throughout.

Formal Description

Convolution (cross-correlation in practice): output element $(i,j)$ is computed as:

${\sum }_{r=0}^{f-1}{\sum }_{c=0}^{f-1}{W}_{rc}\cdot X_{i\cdot s+r,j\cdot s+c}$

Filter $W\in {\mathbb{R}}^{f\times f\times C_{\text{in}}}$ is shared across all spatial positions.

Output size:

$n_{\text{out}}=\lfloor \frac{n+2p-f}{s}\rfloor +1$

where $n$ = input size, $p$ = padding, $f$ = filter size, $s$ = stride.

Padding modes:

• “Valid” convolution: $p=0$, output shrinks
• “Same” convolution: $p=(f-1)/2$, output same size as input (for stride 1)

Multiple output channels: $C_{\text{out}}$ filters each producing one output channel; total parameters = $f\times f\times C_{\text{in}}\times C_{\text{out}}+C_{\text{out}}$ (with biases).

3D input (color images): $X\in {\mathbb{R}}^{H\times W\times C_{\text{in}}}$, filter $\in {\mathbb{R}}^{f\times f\times C_{\text{in}}}$.

Pooling: max pooling takes the maximum over an $f\times f$ window; average pooling takes the mean; typically $f=2,s=2$ halves spatial dimensions; no learnable parameters.

Applications

All CNN-based vision models; feature extraction for images. Modern variants include depthwise separable convolution (MobileNet), dilated convolution, and transposed convolution (upsampling).

Trade-offs

• Parameter sharing assumes translation invariance — fails for tasks with position-specific patterns
• Pooling loses spatial precision (problematic for detection/segmentation, which use feature pyramids instead)
• Max pooling is not differentiable at ties (but in practice this is ignored)

Links

• cnn_architecture
• backpropagation (in 01_foundations/deep_learning_theory)