Three Taps and a Shortcut

Stacking more layers onto a deep plain network made it worse — worse on the training set, which rules out overfitting and leaves an optimisation failure. The fix was one addition: carry a block's input past its layers and add it back, so the block computes $F(x) + x$ instead of $F(x)$.

Here you will run a stack of those blocks over a single row of a feature map, so the whole thing stays small enough to follow by hand.

Task: write residual_stack(signal, kernels), returning the final row as a plain list, rounded to 4 decimal places.

• signal is a list of numbers — one row of a feature map.
• kernels is a list of blocks, one 3-tap filter $[k_0, k_1, k_2]$ each.
• A block takes the row $x$ handed to it and returns a row of the same length:

$F_i = k_0\,x_{i-1} + k_1\,x_i + k_2\,x_{i+1}, \qquad y_i = \max(0,\, F_i) + x_i$

• Anything off either end reads as $0$ — zero padding, so the length never changes.
• Blocks run in order, each one taking the row the previous block produced.
• Round only the values you finally return.

One of the tests hands a block a filter of all zeros. In a plain stack that layer would wipe the row out and everything after it would be working with nothing; here it is forced to pass its input straight through. That is the whole argument for skip connections — a layer with nothing useful to add costs nothing instead of doing damage, which is what stopped extra depth from being a risk.