# Why NNMF (non-negative matrix factorization) is a method for linear dimensionality reduction?

Some sources (for example this) say that NNMF is a method for linear dimensionality reduction. How to prove this statement?

I see two different explanations of this and I want to know which of them (if any) is more correct.

Firstly, using the idea from this post, we can say that NNMF is linear because it assumes that the low-dimensional manifold (on which samples are projected) is linear. I am not sure that such assumption really takes place.

Secondly, we can say that NNMF is linear because it decomposes the original matrix $$X$$ by the product of two matrices $$W H$$. Matrix product is a linear operation, so NNMF is a linear dimensionality reduction method. The similar thing was done in this post, devoted to PCA.

• The second explanation is definitely correct. The first one is a bit sloppy in terms of its wording. Multiplication canbe though as a linear projection. That said, a low dimensional manifold can still be nonlinear (think of standard LLE for example) so that itself doesnot mean much. Feb 24 '21 at 20:47