Build a customer segmentation & recommendation engine

Step through the three stages. The embedding stage doesn't know about "segments" — it just lays out points so similar customers end up close together. The segmentation stage doesn't know about "recommendations" — it just finds two groups in whatever geometry it's handed. Each stage feeds the next one blind.

Three mechanisms, chained: the output of each becomes the input to the next.

• $v_{ij}$ — this part's UMAP fuzzy-graph edge weight, built from the 6 customers' raw 3-feature vectors, then laid out in 2D by attraction and repulsion.
• $P(x)$ — the k-means++ probability of picking $x$ as a centroid, run on the 2D layout instead of the original features.
• $\hat r_{ij}$ — the matrix-factorization prediction for customer $i$'s rating of product $j$, trained independently of the geometry entirely.

1. Embed: 3 features become 2D geometry

   The fuzzy neighbor graph and force layout from this part's UMAP chapter run unmodified on the customers' raw behavioral features — no relabeling, no rewriting, just a direct function call.

2. Segment: geometry becomes group membership

   k-means++ seeding and Lloyd's algorithm, also unmodified, run on the 2D layout instead of the original k-means chapter's synthetic blobs.

3. Recommend: a separate, parallel signal — product ratings

   Matrix factorization trains on a completely different dataset (product ratings, not behavioral features) using the exact same gradient-descent mechanism as the standalone chapter. It never sees the embedding or the segments at all.

With one latent factor, the recommendation stage badly misjudges customers whose taste doesn't reduce to a single number — the same capacity limit the standalone chapter demonstrated on a smaller ratings table. With two, the fit tightens substantially, even though this table isn't perfectly rank-2 like the earlier toy example.

Two behaviorally opposite groups of customers, run through all three stages:

1. Embedding collapses each group to (essentially) one point

   C1, C2, and C3 — all frequent, recent, high-spending — land within $0.0001$ units of each other after layout. C4, C5, C6 do the same, at a different point roughly $27.2$ units away.

2. Segmentation recovers exactly those two groups

   k-means++ seeding on that 2D layout, followed by Lloyd's algorithm, assigns C1-C3 to one segment and C4-C6 to the other — with no access to the original features, only the embedded positions.

3. Recommendation fills in the missing rating using a completely separate signal

   C2's rating of product Y was never collected. Trained on the ratings table alone — with no knowledge of C2's segment — two latent factors predict it at about $3.59$; one factor predicts a noticeably worse $2.79$, the same capacity gap seen in isolation.