Latent Factor Models

A latent factor model maps users and items to hidden coordinates and scores compatibility from those coordinates. In recommender systems, matrix factorization is the canonical example, but factor models can also include side features, biases, context, or neural encoders.

The scoring contract

The core scoring contract is

$$s(u,i)=p_u^{\top }{q}_i,$$

or with biases,

$$s(u,i)=\mu +b_u+b_i+p_u^{\top }{q}_i.$$

The factors are “latent” because no column is pre-labeled as genre, price sensitivity, or expertise. Their meaning comes from the training objective and data. Content-based recommendation starts from observed item features; latent factor models infer features from interaction patterns.

Worked example

For a user vector $p_u=[1.2,-0.4]$, score each item by a dot product:

Item	Item vector $q_i$	Score $p_u^{\top }{q}_i$
0	$[1.0,-0.2]$	$1.2(1.0)-0.4(-0.2)=1.28$
1	$[-0.3,1.1]$	$1.2(-0.3)-0.4(1.1)=-0.80$
2	$[0.8,-0.5]$	$1.2(0.8)-0.4(-0.5)=1.16$

Item 0 is the top recommendation because it has the largest dot product. Item 0 and item 2 align with the user vector; item 1 points in the opposite direction. Hybrid recommenders often combine these latent scores with content features to reduce cold-start damage.

Caveats

Latent dimensions are useful but not guaranteed to be stable or interpretable across retrains. They reflect exposure and feedback loops in the logs, not pure preference. When a product needs explanation, constraints, or editorial control, latent factors usually feed a broader retrieval and ranking architecture rather than serving alone.

References

• Koren, Bell, and Volinsky, 2009, Matrix Factorization Techniques for Recommender Systems
• Rendle, 2012, BPR: Bayesian Personalized Ranking from Implicit Feedback

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