Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!