# How to bootstrap non-negative matrix factorization results?

I have RNA-seq data from 9 samples and around 15,000 genes. I know that these 9 samples consist of varying proportions of two cell types, each with their own expression profile. I am using non-negative matrix factorization with a rank of 2 to determine the cell-type ratios in each sample. In other words, I use NMF to decompose my 9 x 15,000 matrix into a 2 x 15,000 matrix and a 2 x 9 matrix.

I want to obtain confidence intervals for my estimates of the 2 x 9 matrix. From what I understand about bootstrapping, I could do this by resampling from my 9 observations a bunch of times and keep track of each estimate for each observation.

My question is, since NMF doesn't have any notion of observations vs. features, could I also estimate the variance by resampling genes (features) rather than observations?

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The NMF is not, in general, unique, so some or all of the parameters may be unidentifiable. Why should you, then, expect to get confidence intervals for them? –  cardinal Jun 17 '12 at 18:12
It would be polite, especially to new users posting their very first question, for downvotes to provide some constructive feedback on how they think the question could be improved. –  cardinal Jun 17 '12 at 23:42
@cardinal: I can use outside information to map the NMF basis vectors to cell types, preserving identifiability. [this plot] (imgur.com/zvjvn) shows my estimates for cell type 1 for the 9 samples obtained by repeating the NMF 100 times (I know this doesn't give me a real confidence interval, I was looking at how much variability NMF itself has). If I was having identifiability issues, I would expect much greater spreads as the results would randomly alternate between the proportion and 1-proportion. –  Amit Deshwar Jun 19 '12 at 14:27
@AmitDeshwar: please consider selecting one of the answers or explaining why neither satisfies your question. –  jrennie Aug 20 '12 at 12:54

It turns out the answer is yes.
I read Efron and Tibshirani's textbook on the bootstrap and they gave an example of bootstrapping PCA applied to standardized test data where a few students answered many questions. In this example they resampled questions (equivalent to genes in my example). Since NMF is a special case of the more general multinomial PCA, it seems to follow that this approach should also work with NMF.

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NNMF is unique to within a permutation and a scaling, per Laurberg, 2007, 14th IEEE/SP Workshop on Statistical Signal Processing, August 2007. Related, Donoho, Stodden "When does NNMF give a correct decomposition into parts?", Stanford, 2003.

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From Laurberg: "Theorem 8 states that the NMF is unique if the row vectors in W are boundary close and the column vectors in H are sufﬁciently spread". AFAICT, NMF is not generally unique to within permutation/scaling. –  jrennie Aug 15 '12 at 13:31