Before you can read the parent note, you need a toolbox. Below is every symbol and idea it uses, built from nothing, each one leaning on the one before it. Read top to bottom.
The little i below a letter is just an index — a row number, like a seat number. When you see ∑i=1n it means "walk through every seat from 1 to n and add up."
Look at the figure: each row is one example (one cyan dot with its answer yi). Each column is one feature — a property measured for every row. This row/column split is the whole game: rows are examples, columns are features.
xij = the value in row i, column j. Two indices: first says which row, second says which feature.
d = the number of features (columns). So w has one dial for each column.
Why weights? To predict, we take each feature, multiply by its dial, and add them all up:
y^i=w1xi1+w2xi2+⋯+wdxid.
A big wj = "this feature strongly pushes the prediction"; wj=0 = "ignore this feature completely." Deleting a feature just means setting its weight to zero — remember that, it is the heart of sparsity.
Why this tool and not another? A weighted sum is the simplest way to combine many numbers into one prediction where each feature's influence is a single tunable strength. That simplicity is why linear regression uses it — and Elastic Net is linear regression with a penalty bolted on.
We have a guessy^i=w⊤xi and a truthyi. How wrong are we?
Why square it? Two reasons, both visible in the figure below. First, squaring makes every miss positive, so overshoots and undershoots don't cancel out. Second, it punishes big misses far more than small ones (a miss of 2 costs 4, a miss of 3 costs 9), which pushes the line to avoid large blunders.
This is the term labelled "data fit" in the parent note. Small MSE = the line hugs the data.
A "norm" is a way to measure how big a vector is — how far the whole set of weights is from all-zero. Elastic Net uses two different rulers.
Why two rulers, and why do their shapes differ? Draw all weight-pairs (w1,w2) that a ruler rates as "the same size." For L1 you get a diamond (corners on the axes); for L2 you get a circle. Those shapes are the secret to everything, so look carefully:
Why absolute value for L1 specifically? Absolute value has a sharp kink at zero, and that kink is precisely what creates the corner that snaps weights to 0. A smooth ruler (like L2) has no kink, hence no snapping. This is the whole reason the two behave differently.
Why we care: if the whole objective is a bowl, there is one lowest point, and rolling downhill always finds it — no getting stuck in a fake dip. MSE is a bowl, L2 is a strict bowl, L1 is a bowl (with a crease). Their sum is a bowl, so Elastic Net has a well-defined best answer.
