  In a variety of settings, limitations of sensing technologies or other
sampling mechanisms result in missing labels, where the likelihood of a missing
label in the training set is an unknown function of the data. For example,
satellites used to detect forest fires cannot sense fires below a certain size
threshold. In such cases, training datasets consist of positive and
pseudo-negative observations where pseudo-negative observations can be either
true negatives or undetected positives with small magnitudes. We develop a new
methodology and non-convex algorithm P(ositive) U(nlabeled) - O(ccurrence)
M(agnitude) M(ixture) which jointly estimates the occurrence and detection
likelihood of positive samples, utilizing prior knowledge of the detection
mechanism. Our approach uses ideas from positive-unlabeled (PU)-learning and
zero-inflated models that jointly estimate the magnitude and occurrence of
events. We provide conditions under which our model is identifiable and prove
that even though our approach leads to a non-convex objective, any local
minimizer has optimal statistical error (up to a log term) and projected
gradient descent has geometric convergence rates. We demonstrate on both
synthetic data and a California wildfire dataset that our method out-performs
existing state-of-the-art approaches.
