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Generative Adversarial Network (GAN) by hand ✍️ ~ 9 steps walkthrough below
The Gen in GenAI came from this landmark paper by Ian Goodfellow et al., 12 years ago.
The paper showed that a neural network can not only classify but also turn upside down to generate realistic looking images.
The secret? We pit two of them against each other: a Generator turns noise into fake data, and a Discriminator learns to tell fake from real, pushing the Generator to keep doing better.
One runs upside down, the other right way up.
I drew and calculated one entirely by hand.
Goal: generate realistic 4D data out of 2D noise, filling in every cell yourself.
= 1. Given =
Four noise vectors in 2D, and four real data vectors in 4D.
= 2. Generator, first layer =
Let us multiply the noise by weights and biases to get new features.
= 3. ReLU =
We apply the activation, and -1 and -2 are crossed out and set to 0.
= 4. Generator, second layer =
Let us multiply again. ReLU applies here too, but every value is already positive, so nothing changes. What comes out is the fake data F, made by a two-layer generator out of nothing but noise.
= 5. Discriminator, first layer =
We feed it both, the four fakes and the four real vectors, through the same weights. It never learns which is which from the layout, only from the numbers.
= 6. Discriminator, second layer =
Let us reduce each data vector to a single feature Z. Eight vectors in, eight numbers out.
= 7. Sigmoid =
We turn each Z into a probability Y. A 1 means the discriminator is certain the data is real, a 0 means certain it is fake.
= 8. Training the Discriminator =
Let us take the gradients as Y minus YD, where YD is what the discriminator should have said: 0 for the four fakes, 1 for the four real. Why so simple? Because pairing sigmoid with binary cross entropy loss makes the math collapse to exactly this subtraction. Its loss uses both halves of the page.
= 9. Training the Generator =
We do it again, as Y minus YG, and YG is [1, 1, 1, 1]: the generator wants the discriminator to call every fake real. Same predictions, different target, opposite goal. Its loss uses only the fakes.
The outputs:
Fake data F = [1, 2, 3, 1], [1, 1, 2, 1], [2, 2, 4, 2], [1, 0, 1, 1]
Predictions on fakes = [.7, .5, .9, .3]
Predictions on real = [.7, .9, .9, 1]
Discriminator gradients = [.7, .5, .9, .3] and [-.3, -.1, -.1, 0]
Generator gradients = [-.3, -.5, -.1, -.7]
The takeaway: the adversarial part is one subtraction done twice. The same eight predictions, scored against two opposite targets, send one set of gradients back through the blue weights and another back through the green ones.
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U-Net by hand ✍️ ~ 17 steps walkthrough below
I consider U-Net as a key milestone in deep learning, the first image-to-image model that really worked!
It came out of medical imaging, an unusual place, not from NeurIPS or CVPR or ACL.
Now it is the backbone of diffusion models, which you see in almost all modern image generation models.
I drew the network as a C so the matrix multiplication flows naturally down.
Tilt your head to the right and it is a U again. 🤣
Goal: push a 3 x 16 image down to a 2 x 4 bottleneck and back out again, filling in every cell yourself.
= 1. Given =
An image of three channels, R, G and B, sixteen pixels wide, and every kernel the network will use.
= 2. Convolution 1 =
Let us slide the first kernel over the image. Each output is one multiply-and-add over a 2 x 3 window, and the result is the green feature map.
= 3. Find the maxima =
We circle the largest value in each 1 x 2 window. Circling first is worth the extra step: it is the pooling decision, made before anything is written down.
= 4. Max pool 1 =
Let us copy those maxima down. Sixteen columns become eight, and half the detail is gone for good.
= 5. Convolution 2 =
We convolve again with the second kernel, deeper into the contracting path. The feature map is blue now.
= 6. Find the maxima again =
Same move as step 3, on the blue map.
= 7. Max pool 2 =
Eight columns become four.
= 8. The bottleneck =
Let us convolve once more. This is the bottom of the U, a 2 x 4 block that is everything the network kept.
= 9. Spread it out =
We start back up. The transposed convolution writes each bottleneck value into a wider grid, leaving gaps between them.
= 10. Transposed convolution 1 =
Let us fill those gaps by convolving over the spread-out grid. Four columns become eight.
= 11. The first skip =
We copy the encoder's matching row straight across. This is the skip connection, and it is the whole reason a U-Net can recover detail that pooling threw away.
= 12. Convolution with the skip =
Let us convolve the upsampled features together with the copied ones.
= 13. Spread it out again =
Same as step 9, one level up.
= 14. Transposed convolution 2 =
Eight columns become sixteen, back to the width we started at.
= 15. The second skip =
The encoder's first feature map comes across, the one made before any pooling happened.
= 16. Convolution and ReLU =
We convolve, then cross out every negative and set it to zero.
= 17. Output convolution =
Let us apply the last kernel. Out comes R', G' and B', an image the same size as the one we started with.
The outputs:
R' = [3, 0, 7, 0, 7, 0, 17, 0, 3, 0, 9, 0, 2, 0, 6, 0]
G' = [1, 20, 1, 10, 1, 12, 1, 19, 2, 5, 1, 11, 1, 3, 1, 7]
B' = [4, 20, 8, 10, 8, 12, 18, 19, 5, 5, 10, 11, 3, 3, 7, 7]
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