Unsupervised Learning of the Set of Local Maxima

This paper describes a new form of unsupervised learning, whose input is a set of unlabeled points that are assumed to be local maxima of an unknown value function $v$ in an unknown subset of the vector space. Two functions are learned: (i) a set indicator $c$, which is a binary classifier, and (ii) a comparator function $h$ that given two nearby samples, predicts which sample has the higher value of the unknown function $v$. Loss terms are used to ensure that all training samples $\vx$ are a local maxima of $v$, according to $h$ and satisfy $c(\vx)=1$. Therefore, $c$ and $h$ provide training signals to each other: a point $\vx'$ in the vicinity of $\vx$ satisfies $c(\vx)=-1$ or is deemed by $h$ to be lower in value than $\vx$. We present an algorithm, show an example where it is more efficient to use local maxima as an indicator function than to employ conventional classification, and derive a suitable generalization bound. Our experiments show that the method is able to outperform one-class classification algorithms in the task of anomaly detection and also provide an additional signal that is extracted in a completely unsupervised way.