Cross-Domain Recommendation

Lecture context: transferring recommendation knowledge across domains, aided by LLM world knowledge.

Definition

Cross-Domain Recommendation (CDR)

Cross-domain recommendation transfers preference knowledge learned in a source domain ${\mathcal{D}}^S$ (e.g., books) to improve recommendation in a target domain ${\mathcal{D}}^T$ (e.g., movies), typically to combat Data Sparsity and the Cold Start Problem in the target. A “domain” is any partition of items (and sometimes users) by catalogue, platform, or modality, where the two domains share an overlap — overlapping users (the usual case), overlapping items, or shared semantic content.

Classical CDR builds an explicit bridge that maps source-domain user representations into the target-domain space. The generative / LLM era reframes this: a pretrained LLM already encodes cross-domain world knowledge, so transfer can happen implicitly through shared semantics rather than a hand-built mapping.

Intuition

Borrow signal where you have it, spend it where you don't

A new user on a movie service has almost no clicks (cold start), but the same user has a rich history on a books service. Their taste — “likes dark, character-driven narratives” — is domain-invariant. If we can express that latent taste once and project it into the movie space, we recommend well from day one.

The hard part is the semantic gap: a book embedding and a movie embedding live in different spaces with different ID vocabularies, so you cannot just reuse the vector. Three escalating answers:

1. Learn a bridge $f^{S\to T}$ that maps source user factors to target user factors (classic embedding-and-mapping CDR).
2. Share a backbone so both domains are trained jointly and the representation is forced to be shared.
3. Use language as the universal interface — describe users and items in text, and let an LLM’s world knowledge supply the cross-domain prior for free. This is why the lecture lists cross-domain transfer as a core advantage of Generative Recommendation.

Mathematical Formulation

The canonical embedding-and-mapping formulation (e.g., the EMCDR scheme): learn user/item latent factors separately in each domain via Matrix Factorization, then learn a mapping that aligns the spaces using the overlapping users ${\mathcal{U}}^{S\cap T}$ as supervision.

$${\mathcal{L}}_{\text{map}}=\sum _{u\in {\mathcal{U}}^{S\cap T}}\Vert f_{\theta }\left({\mathbf{p}}_u^S\right)-{\mathbf{p}}_u^T{\Vert }_2^2{\hat{r}}_{u,i}^T=f_{\theta }({\mathbf{p}}_u^S{)}^{\top }{\mathbf{q}}_i^T$$

where:

• ${\mathbf{p}}_u^S\in {\mathbb{R}}^d$ — user $u$‘s latent factor learned in the source domain
• ${\mathbf{p}}_u^T\in {\mathbb{R}}^d$ — user $u$‘s latent factor learned in the target domain (only available for overlapping users)
• ${\mathbf{q}}_i^T\in {\mathbb{R}}^d$ — target-domain item $i$‘s latent factor
• $f_{\theta }:{\mathbb{R}}^d\to {\mathbb{R}}^d$ — the bridge / mapping network (linear or MLP), trained on overlapping users
• ${\mathcal{U}}^{S\cap T}$ — set of users present in both domains (the supervision signal)
• ${\hat{r}}_{u,i}^T$ — predicted target-domain score for a cold-start user, obtained by mapping their source factor into the target space

At inference, a user with no target-domain history but a known source factor ${\mathbf{p}}_u^S$ is scored entirely through $f_{\theta }({\mathbf{p}}_u^S)$ — knowledge has been transferred across the domain boundary.

LLM-based transfer (the lecture’s framing). Instead of an explicit $f_{\theta }$, both domains are verbalized into a shared text space and a single LLM scores or generates:

$${\hat{y}}_u^T=\text{LLM}(\text{prompt}(h_u^S,h_u^T,c^T))$$

where $h_u^S,h_u^T$ are the user’s natural-language histories in each domain and $c^T$ describes the target item/candidate. The cross-domain prior is no longer learned from overlap data — it is baked into the pretrained weights (e.g., the model already “knows” that fans of a certain author enjoy a certain director). The mapping $f_{\theta }$ is replaced by the LLM’s internal world knowledge.

Key Properties / Variants

• Overlap regimes: user-overlap (most common; enables EMCDR-style bridges), item-overlap, and fully non-overlapping (hardest — requires content/semantic anchors, exactly where LLMs help).
• Direction: single-direction transfer (${\mathcal{D}}^S\to {\mathcal{D}}^T$) vs. dual/joint transfer where both domains improve each other.
• What gets transferred: (i) latent factors via a bridge (EMCDR); (ii) a shared embedding space trained jointly; (iii) semantic/world knowledge via text or a shared codebook of Semantic IDs.
• Primary motivation: mitigate Data Sparsity and target-domain Cold Start — the slide explicitly pairs cross-domain transfer with cold-start as the regime where generative beats discriminative.
• Discriminative limitation: a Collaborative Filtering scorer $f(\text{user},\text{item})$ is tied to a fixed candidate pool in one domain and cannot reason about an unseen domain’s items; generation over a shared semantic vocabulary can.
• LLM-as-Recommender route: zero-shot prompting transfers across scenarios with no fine-tuning, because the model’s pretraining already spans domains; the cost is prompt sensitivity and item hallucination (it may “recommend” a target item that does not exist), motivating generation grounding.
• LLM-as-Enhancer route: the LLM rewrites cross-domain histories into enriched text features that a downstream Recommender System consumes — knowledge transfer as feature augmentation.

Classic embedding-and-mapping pipeline:

Algorithm: Embedding-and-Mapping CDR (EMCDR-style)
──────────────────────────────────────────────────
Input: source interactions R^S, target interactions R^T,
       overlapping users U^{S∩T}
 
1. Learn source factors  {p^S_u}, {q^S_i}  via MF on R^S
2. Learn target factors  {p^T_u}, {q^T_i}  via MF on R^T
3. Train bridge f_θ on overlapping users:
     minimize  Σ_{u∈U^{S∩T}} || f_θ(p^S_u) − p^T_u ||²
4. For a cold-start target user u (has p^S_u, no p^T_u):
     p̂^T_u ← f_θ(p^S_u)
     score item i:  r̂^T_{u,i} ← (p̂^T_u)ᵀ q^T_i
5. Recommend Top-K items by r̂^T_{u,i}

Negative transfer and overlap dependence

Transfer is not always helpful. If domains share little true preference structure, forcing a shared space causes negative transfer (source noise degrades the target). Classic bridges also depend on a sufficiently large overlapping-user set ${\mathcal{U}}^{S\cap T}$ for supervision — when overlap is tiny or absent, embedding-and-mapping collapses, which is the gap content-based and LLM-based methods fill. LLM transfer dodges the overlap requirement but inherits hallucination and prompt sensitivity, and its world knowledge can be stale or biased for niche/long-tail catalogues.

Connections

• Mitigates: Data Sparsity, Cold Start Problem, Long Tail
• Built on: Matrix Factorization, Collaborative Filtering, Content-Based Recommendation
• Enabled in the generative era by: Generative Recommendation, LLM-based Recommendation, LLM-as-Enhancer, LLM-as-Recommender
• Shared vocabulary mechanism: Semantic IDs, Item Tokenization
• Contrasts with: discriminative single-domain scoring (Top-K Recommendation over a fixed pool)
• Knowledge source: Large Language Models (LLM) world knowledge; complements Hybrid Recommendation

Appears In

• RS-L03b - From LLMs to LRMs