Neighbourhood Methods

#Item-based kNN

sim(i, j)   similarity of item columns over users
N_k(i)      the k most similar items to i
score(u, i) = sum over j in history(u) with i in N_k(j) of sim(j, i) * w(u, j)

Item-based kNN is more stable than user-based kNN: items accumulate many interactions, while users often have few. Similarities can be precomputed offline, and explanations come for free (the history item that contributed most).

#Similarities

#Shrinkage

sim' = sim * n_ij / (n_ij + lambda), where n_ij is the number of common users. Without it, two items seen together by one user can get similarity 1. Typical lambda is 10-100, tuned on validation. Dense data needs less (the fixture preferred 0).

#Neighbourhood size

Small k (10-20) gives precise but less diverse lists. Large k (100-500) behaves more like popularity within taste clusters. Tune k together with shrinkage.

#User-based kNN

score(u, i) = sum over v in N_k(u) of sim(u, v) * r(v, i). Useful for very small catalogues or when users are stable and rich in history. Otherwise item-based kNN is preferred.

#Explanations

Report the history item with the largest contribution: "Because you liked Dune". itemknn.py writes it in the because column, or popular for fill-ins. Explanations increase trust and help debugging: nonsensical "because" items reveal noisy signals.

#Libraries

# scikit-learn for similarities on a sparse matrix
from sklearn.metrics.pairwise import cosine_similarity
S = cosine_similarity(X_item_user, dense_output=False)          # X: items x users (csr)
# implicit's kNN
from implicit.nearest_neighbours import CosineRecommender, BM25Recommender
model = BM25Recommender(K=100); model.fit(user_items_csr)
ids, scores = model.recommend(user_id, user_items_csr[user_id], N=10)

BM25 weighting of the interaction matrix often beats plain cosine on implicit data.