Abstract

Consider the problem of classifying a sample X₀ into one of two classes, using a training set Q. Let Q be composed of l labeled samples {(X₁, θ₁), …, (X_l, θ_l)} and u unlabeled samples {X′₁, …, X′_u}, where the labels θ_i are i.i.d. Bernoulli(η) random variables over the set {1, 2}, the observations {X_i}_i=1^l are distributed according to f_θi(·) and the unlabeled observations {X′_j}_j=1^u are independently distributed according to the mixture density f_X′(·) = ηf₁(·) + (1−η)f₂(·). We assume that f₁(·),f₂(·) and η are all unknown. Let f₁(·) and f₂(·) belong to a known family , and assume that the mixtures of elements of are identifiable. Even when the number of unlabeled samples is infinite and the decision regions can therefore be identified, one still needs labeled samples to label the decision regions with the correct classification. Letting R(l, u) denote the optimal probability of error for l labeled andu unlabeled samples, and assuming that the pairwise mixtures of are identifiable, we obtain the obvious statements

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