What it means

• Each item (movie, product, song, paper, etc.) is represented by a feature vector: genres/tags, text keywords, image or audio embeddings, price/brand indicators, etc.
• Cosine similarity measures how aligned two item vectors are—i.e., how similar their feature profiles are—independent of scale (length).

The math

For item vectors $x, y \in \mathbb{R}^d$:

$\text{cosine\_sim}(x, y)=\frac{x\cdot y}{\lVert x\rVert_2\,\lVert y\rVert_2} =\frac{\sum_{k=1}^d x_k y_k}{\sqrt{\sum_{k=1}^d x_k^2}\,\sqrt{\sum_{k=1}^d y_k^2}}$

• Range is $[-1, 1]$ in general; with non‑negative features (e.g., one‑hot, TF‑IDF), it’s typically $[0,1]$.
• Cosine distance (often used in diversity): $1 – \text{cosine\_sim}(x,y)$.

Why cosine?

• Scale invariance: If you double all features for an item, direction stays the same, so similarity doesn’t change. That’s ideal when only relative composition matters (e.g., genre mix, term proportions).
• Works well with sparse, high‑dimensional vectors (bags of words, tags).
• Fast to compute and index; compatible with approximate nearest‑neighbor search (ANN).

Building item feature vectors

Common ways to encode items:

1. One‑hot / multi‑hot categories (genres, brands, tags).
2. TF‑IDF from text fields (title/description → token weights).
3. Learned embeddings (e.g., sentence transformers for text, CNN embeddings for images, audio embeddings).
4. Hybrid vectors (concatenate normalized content features + numeric attributes; consider scaling/standardization first).

Worked mini‑example

Suppose a simple genre vector: $[{\small \text{Action},\ \text{Romance},\ \text{Comedy}}]$

• Movie A: $x=[1,0,1]$ (Action & Comedy)
• Movie B: $y=[1,0,0]$ (Action only)

$x\cdot y = 1,\quad \lVert x\rVert=\sqrt{2},\ \lVert y\rVert=1 \Rightarrow \text{cosine\_sim}(x,y)=\frac{1}{\sqrt{2}}\approx 0.707$

They’re fairly similar because both are Action; A has extra Comedy signal.

How it’s used in recommender systems

• Item‑to‑item similarity: “Because you liked X, here are items with high cosine similarity to X’s features.”
• Cold‑start content‑based recs: When a new item has no interactions yet, use its content features to find neighbors via cosine.
• Diversity (ILD): Define $d(i,j)=1-\text{cosine\_sim}(x_i,x_j)$; then Intra‑List Diversity is the average pairwise $d(i,j)$ across a user’s list.
• Re‑ranking: Start from a relevance shortlist, then re‑rank to maximize relevance + (1 − cosine) to avoid near‑duplicates.

Practical tips

• L2‑normalize feature vectors before computing dot products (or use a library’s cosine function, which effectively normalizes).
• Feature scaling matters: When mixing modalities (text + categories + numbers), standardize/weight sub-blocks so one source doesn’t dominate.
• Zero vectors: Handle items with no features (avoid divide‑by‑zero; drop or impute).
• Dimensionality & sparsity: For huge catalogs, precompute top‑K neighbors or use ANN (e.g., FAISS, ScaNN, HNSW).
• Domain choice of features: Cosine is only as good as the representation; prefer semantically meaningful embeddings/encodings.

Common pitfalls & variants

• Popularity leakage: If features include popularity counts, they can overshadow content; consider log‑scaling or excluding them from cosine.
• Negative entries: Some embeddings are mean‑centered → cosine can be negative; that’s fine, just interpret negative as “anti‑similar.”
• Adjusted cosine (ratings CF): Different concept: compute cosine between item rating vectors after subtracting user means; that’s collaborative, not content‑feature cosine.
• Angular distance: $\theta=\arccos(\text{cosine\_sim})$ is a true metric on the unit sphere; sometimes used in ANN.