Talk:Non-negative matrix factorization

Non-negative matrix approximation
I removed the section about non-negative matrix approximation because there is very little reference to it in the literature (I could only 1 paper linking to this idea). It seems premature to include reference to it at the moment, and its inclusion does not add significantly to the article. Feel free to revert if I am mistaken --Josephus78 04:18, 22 June 2006 (UTC)
 * As you I have only seen the phrase "non-negative matrix approximation" in a single paper, but all the practical applications of NMF that I have seen have been where NMF were approximating the X matrix. -- fnielsen 19:54, 30 June 2006 (UTC)

Equivalences
Did a few edits today. I am a bit confused by the bit about K-means clustering. The formulation and reference [9] seem to imply an equivalence between K-means and NMF. However, in the standard formulation, it can't be so as K-means has no positivity constraints (cf. Lee&Seung's Nature paper). So you need to either relax NMF constraints or add constraints to K-means. Hopefully someone can either correct or clarify the formulation. Sunny house 16:01, 15 March 2007 (UTC)

Also it may not be necessary to keep 2 refs on the NMF-PLSA equivalence. Don't want to offend anybody but it seems the earlier ref. should be enough. Sunny house 16:03, 15 March 2007 (UTC)
 * I have been a bit confused too. Please clean up if you like with the K-means and the NMF-PLSA. &mdash; fnielsen 20:38, 15 March 2007 (UTC)

Reference [15] does not claim that NNMF with the orthogonality constraint on H is equivalent to k-means clustering. It appears to me that they are not equivalent, although close. I think this needs to be corrected. I have not found a good reference. If it is not found, then this claim is better be removed at all. 79.183.10.203 (talk) 14:09, 24 June 2019 (UTC)Baruch Youssin

Minimizing Divergence
In the section Different cost functions and regularizations it says Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules.

This is not true according to:

Amy N. Langville, Michael W. Berry, Murray Browne, V. Paul Pauca, and Robert J. Plemmons. A Survey of Algorithms and Applications for the Nonnegative Matrix Factorization. Computational Statistics and Data Analysis. Elsevier. Submitted Jan. 2006. »Lee and Seung used the gradient and properties of continual descent (more precisely, continual nonincrease) to claim that the above algorithm converges to a local minimum, which was later shown to be incorrect (Chu et al., 2004; Finesso and Spreij, 2004; Gonzalez and Zhang, 2005; Lin, 2005b). In fact, the proof by Lee and Seung merely shows a continual descent property, which does not preclude descent to a saddle point.«

I think this should be corrected. --JohKar (talk) 08:04, 7 December 2010 (UTC)

Ranks
I find the following sentence unclear:

When multiplying matrices the factor matrices can be of significantly lower rank than the product matrix and it's this property that forms the basis of NMF. If we can factorize a matrix into factors of significantly lower rank than the original matrix then the column vectors of the first factor matrix can be considered as spanning vectors of the vector space defined by the original matrix.

The rank of the product of two matrices is lower than the minimal rank among the factors, thus the factor matrices cannot have significantly lower rank than the product matrix. — Preceding unsigned comment added by Mickaël Poussevin (talk • contribs) 08:22, 8 January 2013 (UTC)

Agree with this because it makes unfounded assumptions about linearity. Given that NMF is about dimensionality reduction perhaps the phrase should read: When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it's this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. — Preceding unsigned comment added by Roybgardner (talk • contribs) 11:02, 18 November 2013 (UTC)

Proposed merge with Online NMF
The article Online NMF discusses NMF at some length, but its content regarding online optimization algorithms can be summarized in a single paragraph. Q VVERTYVS (hm?) 15:38, 21 October 2013 (UTC)
 * Support This seems reasonable to me. We have plenty of algorithm articles that have sections on online variants and the online-specific content of the Online NMF article is pretty modest at this point. --Mark viking (talk) 00:14, 26 September 2014 (UTC)
 * Done. Thanks for proposing. --mcld (talk) 20:08, 8 April 2015 (UTC)

Missing values (imputation, null values, None, ...)
I miss a section about missing values (NULL values, None, data imputation). --MartinThoma (talk) 09:06, 22 July 2019 (UTC)