Each categorical feature value (a user id, an item id, a device, a time slot) gets a learned scalar weight wi and a latent embedding vector vi ∈ ℝ² — plotted here directly as its (x,y) position on the plane, with no 3D-to-2D projection needed. A Factorization Machine (Rendle, 2010 — the mechanism behind Wide & Deep and DeepFM production recommenders) scores an example x by combining two terms:
ŷ(x) = w0 + Σᵢ wᵢxᵢ + Σᵢ<ⱼ ⟨vᵢ,vⱼ⟩ xᵢxⱼ
\_____ wide ____/ \_____ FM (deep) _____/
p = sigmoid(ŷ) loss = −[y·log p + (1−y)·log(1−p)]
The wide term (shown as the bar chart) memorizes how good each individual feature value is on its own (linear). The FM term captures pairwise interactions — e.g. "Bob + Mobile" might click more than either alone predicts — as the 2D dot product between the two features' embeddings, which is exactly why the lines connecting active dots on the plane turn green (reinforcing) or red (cancelling) as training moves the points around.
- Training example — pick one value per field (User/Item/Device/Time); only those four dots light up and pair up into 6 interaction lines.
- Label — tell the model whether this example was actually clicked.
- Train one SGD step — a real gradient-descent update: ∂L/∂ŷ = p−y, then wᵢ and vᵢ of the four active features move by −η·gradient; everything else is untouched, since inactive xᵢ=0 zeroes their gradient exactly as in a real FM.
- Auto-train — repeats that same SGD step automatically at the speed you set, so you can watch embeddings converge live instead of clicking repeatedly.
- Score view — inspect what the wide term alone, the FM term alone, or their sum would predict.
- Drag / scroll the plane — pan and zoom the embedding view; it's pure camera movement, the model underneath is unaffected.
Train the same example with y=1 a few times and watch its embeddings pull together (their dot product rises, the loss sparkline falls); train it with y=0 and watch them push apart — that pull/push on latent vectors is literally how Spotify, YouTube and e-commerce CTR models learn which feature combinations to trust.