Unsupervised Learning of the Set of Local Maxima

A BSTRACT 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 x are a local maxima of v, according to h and satisfy c( x) 1 . Therefore, c and h provide training signals to each other: a point x null in the vicinity of x satisfies c (x) 1 or is deemed by h to be lower in value than x . 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. 1 I NTRODUCTION ...from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. We do not observe the even larger quantity of less spectacular forms and we cannot see those forms that are incompatible with existence. In other words, each sample we observe is the result of optimizing some fitness or value function under a set of constraints: the alternative, lower-value, samples are removed and the samples that do not satisfy the constraints are also missing. The same principle also holds at the sub-cellular level. For example, a gene can have many forms. Some of them are completely synonymous, while others are viable alternatives. The gene forms that become most frequent are those which are not only viable, but which also minimize the energetic cost of their expression (Farkas et al., 2018). For example, the genes that encode proteins comprised of amino acids of higher availability or that require lower expression levels to achieve the same outcome have an advantage.