# Interpretation of matrix factorization results

Matrix factorization methods are known to give good results pertaining to problems like movie recommendation. The method reduces the feature space, which is then used for recommendations.

For example consider user item matrix where each element is rating by a user for a product, whose dimension is (lets say) 1000 by 20000. We can apply matrix factorization to this matrix with latent feature size=10. This will result in user latent feature matrix P of size 1000 by 10, and item latent feature matrix Q of size 10 by 20000. Each row of P would represent the strength of the associations between a user and the features. Similarly, each row of Q would represent the strength of the associations between an item and the features.

How do we interpret this reduced latent feature space? What is the relation between reduced latent feature space and actual feature space?

• Hi, Can you explain your question more. people might not be familiar with some terms. Then asking a question without explaining the aspects or some example is difficult to be solved. Thanks. – TPArrow Jun 26 '15 at 8:24
• The latent space is simply the range of latent variables in the general statistical sense: en.wikipedia.org/wiki/Latent_variable. The latent features may or may not correspond to intuitve factors (like genre for movies). – sandris Jul 16 '15 at 8:30

## 1 Answer

Matrix factorization is widely used for its scalability, and its ability to handle sparse datasets, precisely by reducing the feature-space to a smaller, lower-dimensional latent feature-space.

But one of its major drawbacks is the lack of interpretability, because the factorization does not preserve the features that were input, nor is there an easy transformation that helps interpret the latent features. But it also means that it is possibly able to find correlations between features that we would not have thought of, which are somehow more expressive of the preferences.

The closest thing to interpreting the latent features that I have come across, is an idea called Representative Users, where the latent features are seen as users that can be used to represent all other users.

• Not sure what you meant here: "nor is there an easy transformation that helps interpret the latent features" There's a linear mapping from the observed to the latent space and vice versa, which is about as simple as one could hope for among the space of possible mappings. Could you elaborate? – user20160 Jan 1 '18 at 22:19
• @user20160 I was referring to mapping the latent factors to intuitive factors, in order to "interpret" them. Moreover, matrix factorization is a non-convex optimization problem, and not linear. – Antimony Jan 9 '18 at 18:38
• The fact that you call matrix factorization nonlinear makes me wonder if we're talking about the same thing. I'm thinking of methods where the original matrix is factored into a product of matrices (e.g. SVD, PCA, NMF), which implies a linear mapping between the input space and the latent space. What's the nonlinearity you mentioned? – user20160 Jan 10 '18 at 1:01
• You're likely talking about the general problem of matrix factorization, while I am talking about factorizing large, high-sparsity matrices as in the case of recommender systems. One CAN do it using methods like SVD, where you just fill in the missing values with something (for eg. 0) but the objective functions typically used are non-linear and non-convex in nature (for eg. gradient descent, alternate least squares). – Antimony Jan 10 '18 at 21:12