# #StackBounty: #machine-learning #mathematical-statistics Maximizing AUC based on point cloud distance

### Bounty: 50

Let \$V\$ be an \$n\$ dimensional space with sets of positive class vectors \$P\$ and negative class vectors \$N\$. The task is to find a vector \$x\$ such that AUC is maximized, based on ranking generated by computing distances between \$x\$ and \$P,V\$. So in a sense, \$x\$ is closer to \$P\$ than to \$V\$. It looks like this doesn’t have a unique solution, but I’m curious if there is a really easy explicit solution to this, or a short algorithm? Or is this NP-hard?

Surely this is a well known classical problem? One algorithm that I think works is to sample triplets \$(x,p,n)\$ with one positive and one negative vector and then formulate the usual triplet loss:

\$\$L(x,p,n) = max(0,|x-p|^2-|x-n|^2+epsilon),\$\$

which pushes \$x\$ closer to \$p\$ and further from \$n\$. I’m just hoping for something easier.

Get this bounty!!!

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