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.

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.

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

This setup leads one to think of a subset that is 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.

And vaguely this is reminiscent of a distance (or covariance) matrix. Except the confusion matrix is asymmetric but a distance/covariance matrix is not.

Is there a 'meaningful' (coherent, not totally crazy) mapping from a doubly stochastic matrix to a distance matrix? (where a distance matrix is symmetric, non-negative, obeys triangle inequality).

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.

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.