factorization

Technically, any product form is technically a "factorization", but I/we usually reserve the term for structure-based simplifications, described below.

Factorization

Factorization or decomposition is the process of breaking down a complex object (e.g. a function, matrix, distribution, etc.) into a product of simpler objects (factors) that when combined reproduce the original object.

See also matrix factorization.

Factorization of multivariate distributions

If you have a multivariate distribution with independent underlying distributions, then you can factorize these underlying distributions.

$$p(x_1,x_2,\dots ,x_n)=p(x_1)p(x_2)\dots p(x_n)$$

E.g. how much a certain car color is liked and how much a certain car model is liked: $p(\text{model, color})=p(\text{model})\cdot p(\text{color})$

This is a special case of the chain rule of probability.

factorized representations enable compositional generalization