What are mathematically the factors of variation in deep learning?

The following paragraph from an answer tells us about factors of variation

Factors of variation are some factors which determine varieties in observed data. If that factors change, the behaviour of the data will change.

The only interpretation I can do on the bold portion is discriminative features or random variables with high variance.

Is it a correct interpretation? What can be an exact mathematically precise definition for the "factors of variation" if we consider the dataset as a probability distribution?