Consider the non-negative factorization of a positive, real symmetric matrix
A. Non-negative factorization of this matrix yields
A = WH where
W != H^T.
Yes, there are special cases of
A = WW^T factorization, but I understand that this is an imposed constraint. Thus, my question is two-fold:
- Intuitively, why does NMF for a symmetric
A = WW^T?
- What is the meaning of non-transpose-identity in
Ais symmetric? How can the orthogonal factors in
Hbe interpreted in terms of relationships in
A? Is assignment of a loading to
Ha stochastic process? Thus, can conclusions be drawn about the relationships between columns across factors within
My application: I'm trying to decompose gene associations from a gene adjacency matrix. I want to not only decompose the largest sources of variation (i.e. SVD) but also orthogonal and nested gene associations. NMF is ideal for decomposing orthogonal associations as latent factors, and then a similarity generated from relationships across NMF latent factors could enable decomposition of first-order nested gene associations. This sets up an iterative framework for decomposition of higher-order nested gene associations. However, I worry that NMF will randomly partition genetic interaction information due to asymmetry of orthogonal matrices, thus resulting in exponential fragmentation of information with each iteration. I'm at a loss for how to prove or disprove this notion.