1. Definition

  • Intra-List Diversity (ILD) measures how different the items are from each other within a single recommendation list for a user.
  • Goal: Avoid lists where all items are very similar (e.g., 10 action movies that are nearly identical).
  • A good recommender balances relevance (accuracy) with diversity (variety).

2. Formula

Let:

  • $L = \{i_1, i_2, \dots, i_n\}$ be the list of $n$ recommended items.
  • $d(i,j)$ = a distance (or dissimilarity) function between items $i$ and $j$.

Then:

$ILD(L) = \frac{2}{n(n-1)} \sum_{1 \leq p < q \leq n} d(i_p, i_q)$

  • Average pairwise distance between all items in the list.
  • Normalized so ILD ∈ [0,1] if $d(i,j)$ is bounded.

3. Choice of Distance $d(i,j)$

ILD depends on how you define similarity/distance between items:

  • Content-based: cosine distance of feature vectors (e.g., genres, tags, embeddings).
  • Collaborative filtering–based: distance in latent factor space (e.g., matrix factorization embeddings).
  • Taxonomy-based: semantic distance in a category hierarchy.

Typical choice:

$d(i,j) = 1 – \text{sim}(i,j)$

where sim can be cosine similarity.

4. Example

Suppose Top-3 movie recommendations:

  • $i_1 = $ The Avengers (Action)
  • $i_2 = $ Iron Man (Action)
  • $i_3 = $ La La Land (Romance)

If cosine similarity (based on genre vectors) gives:

  • sim(Avengers, Iron Man) = 0.9 → d = 0.1
  • sim(Avengers, La La Land) = 0.2 → d = 0.8
  • sim(Iron Man, La La Land) = 0.3 → d = 0.7

Then:

$ILD = \frac{2}{3(3-1)} (0.1 + 0.8 + 0.7) = \frac{1}{3}(1.6) = 0.53$

So ILD = 0.53 (moderately diverse).

5. Interpretation

  • High ILD → recommendations are very different from each other.
  • Low ILD → recommendations are clustered (lack of variety).

6. Why ILD Matters

  • Improves user satisfaction (not boring or repetitive).
  • Encourages serendipity and discovery of new items.
  • Avoids “popularity bias” where all users get the same narrow set of items.

7. Trade-offs

  • Accuracy vs Diversity:
    • High accuracy models may recommend very similar items.
    • Increasing ILD may lower accuracy slightly but improve novelty.
  • Some approaches:
    • Re-ranking: Start with accurate list, then adjust to maximize diversity.
    • MMR (Maximal Marginal Relevance): Combines relevance and diversity.

Summary:
Intra-List Diversity (ILD) = average pairwise dissimilarity among items in a recommendation list. It evaluates how varied the recommendations are, encouraging systems to balance relevance with diversity so users don’t just see near-identical items.