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I have a similarity matrix $A \in \mathbb{R}^{N\times N}$ and $a_{ij}\ge 0$ and $A$ is also symmetric.

I want to normalize this matrix in order to use it for graph-based clustering, so that each $1 \ge \hat{a}_{ij}\ge 0$. Ideally i like to use this new matrix as a kernel too. How should i normalize it?

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    $\begingroup$ What is the distribution of the similarity scores? If there aren't too many extreme values, you may be able to just max scale it. If there is a long tail, logging may help as well. Not sure about the later part of your question (using transformed matrix as a kernel). $\endgroup$
    – Keith Hughitt
    Oct 20, 2016 at 13:30
  • $\begingroup$ Also it is good to consider whether you really need every value to be in the range 0 to 1. Is it not enough for the variance in indices to be 1? Also, I would look into dividing each entry by the determinant of the matrix. $\endgroup$
    – dimpol
    Oct 20, 2016 at 13:38

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Assuming it's composed solely of positive values, and if your diagonal isn't already composed solely of ones, do:

$$A_{ij}:=\frac{A_{ij}}{\sqrt{A_{jj}\cdot A_{ii}}}$$

This is analogous to the transformation from a covariance to correlation matrix, i.e. diagonals become one, off-diagonal is rescaled.

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