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I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by respective proportions $a_{ki}$ ($i = 1 \cdots N$, $N$ is the disease type):

$$ x_{jk} = \sum_{i=1}^{N} a_{ki}s_{ij} $$

Generally this can be expressed as matrix form

$$ X = AS $$

What I want to do is to solve this equation

\begin{align} &\min_{A}\ || AS- X||^{2}\\ &\text{subject to } \left\{ \begin{array}{c l} \sum_{i} a_{ki} = 1\\ a_{ki} \ge0, \forall i \end{array}\right. \end{align}

Typically this is solved by using quadratic programming. The number of genes is large (e.g. ~30K).

I'm wandering if there is any good probabilistic model to deal such kind of problem?

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  • $\begingroup$ I cannot provide an answer, but your problem sounds like it may be Convex non-negative matrix factorization. If so, you might try searching on that term, or putting it in your question title and/or description. $\endgroup$ – GeoMatt22 Oct 27 '15 at 13:38
  • $\begingroup$ @GeoMatt22: Note quite. In NMF both A and S is unknown. In my problem A is known. $\endgroup$ – neversaint Oct 27 '15 at 13:52
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    $\begingroup$ You mean S is known? (A is the argument of the objective function) $\endgroup$ – GeoMatt22 Oct 27 '15 at 14:02
  • $\begingroup$ @GeoMatt22: Oops you're right. S is known. $\endgroup$ – neversaint Oct 28 '15 at 0:30

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