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I'm reading through this paper: Baran, Y., Bercovich, A., Sebe-Pedros, A., Lubling, Y., Giladi, A., Chomsky, E., ... & Tanay, A. (2019). MetaCell: analysis of single-cell RNA-seq data using K-nn graph partitions. Genome biology, 20(1), 1-19.

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Maybe I am not familiar with this notation but I don't understand the regularization procedure as follows:

  1. What is happening in these max functions? What does the 0 indicate? They have some terms $x$ and then in the max function they write $max( x, 0)$.
  2. I assume $K$ is some regularization parameter, like a cost. Is $s_{ij} * s_{ji}$ the rank product? I.e. the geometric mean?
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Without looking in too much detail:

  • We may suppose that $S$ is a matrix with dimensions indexed as $(i, j)$. So $s_{ij}$ is the cell of $S$ indexed to the $i$th row and $j$th column, and $s_{ji}$ is the cell vice versa. Therefore, symmetrizing $S$ is a matter of ensuring that $s_{ij} == s_{ji}$, which you can achieve by multiplying the (ranked?) entries. Symmetric matrices are nice for doing certain kinds of math, and yield a handy representation of undirected graphs, which the authors appear to be using for kNN.
  • $max(x, 0)$ is just saying that you want either $x$ if $x$ is non-negative, or $0$ if it is negative. Thus, you make the whole matrix non-negative, which makes some handy matrix math possible. You can see how this works computationally by inspecting e.g. the Python max() function.
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