Explain how convolutional layers work and why they are well-suited to image data.

A convolutional layer applies a set of learnable filters (kernels) by sliding each filter over the spatial dimensions of the input and computing a dot product at each position. For a 2D image, a kernel of size k×k with C_in input channels and C_out output channels has k×k×C_in×C_out parameters total. This produces one feature map per output channel, where each value represents the response of that filter at a specific spatial location.

CNNs are powerful for images because of two structural inductive biases they encode: (1) translation equivariance — the same filter is applied everywhere, so if an object moves in the image, the corresponding feature map activation moves identically; (2) parameter sharing — instead of a separate weight per input-output pixel pair (as a fully-connected layer would require), the filter weights are shared across all spatial locations, drastically reducing parameters and improving sample efficiency.

import torch
import torch.nn as nn

# Standard Conv2d usage
# Input:  (batch, C_in, H, W)
# Output: (batch, C_out, H', W')
conv = nn.Conv2d(
    in_channels=3,    # RGB image
    out_channels=64,  # 64 filters
    kernel_size=3,    # 3x3 kernel
    stride=1,
    padding=1,        # 'same' padding — preserves H and W
)

x = torch.randn(8, 3, 32, 32)   # batch of 8 RGB 32x32 images
out = conv(x)
print(out.shape)  # torch.Size([8, 64, 32, 32])

n_params = 3 * 64 * 3 * 3 + 64  # weights + biases
print('Parameters:', n_params)   # 1792
# Compare: FC layer 3*32*32 -> 64*32*32 would be 3*32*32*64*32*32 = 603M!

# Typical CNN block:
cnn_block = nn.Sequential(
    nn.Conv2d(64, 128, kernel_size=3, padding=1),
    nn.BatchNorm2d(128),
    nn.ReLU(),
    nn.MaxPool2d(kernel_size=2, stride=2),  # halves spatial dims
)