Suppose I have a confusion matrix $A$ for a set of points: entry $i,j$ is the fraction of time over all $j$ (and similarly over all $i$) that when $i$ is present then $j$ is recognized (This means doubly stochastic. I don't think there are any other mathematical restrictions.).

This setup (from psychometric data about stimuli that are miscoded) leads one to think of a subset that has elements that are mutually more confusable within the subset than between pairs of elements where one is inside and the other outside the subset.

This is vaguely reminiscent of a distance (or covariance) matrix where a subset may have mutually small distances but distances outside the subset are larger.

Except, among other substantive mathematical differences, a distance/covariance matrix is symmetric but a confusion matrix is in general not.

Is there a 'meaningful' (coherent, not totally crazy) mapping from a doubly stochastic matrix to a distance matrix? That somehow preserves a vague notion of clustering?

This is the application motivation for the question in Distance measure on weighted directed graphs. I think they are separable questions, but they can inform each other.

doublystochastic? If that's an additional constraint you're imposing, be clear about that. But as-is, you seem to allow one particular point j to be recognized 100% of the time, in which case column j has all 1s and everything else is 0. $\endgroup$