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Minimizing error in prevalence estimates and diagnostic classifiers remains achallenging task in serology. In theory, these problems can be reduced tomodeling class-conditional probability densities (PDFs) of measurementoutcomes, which control all downstream analyses. However, this task quicklysuccumbs to the curse of dimensionality, even for assay outputs with only a fewdimensions (e.g. target antigens). To address this problem, we propose atechnique that uses empirical training data to classify samples and estimateprevalence in arbitrary dimension without direct access to the conditionalPDFs. We motivate this method via a lemma that relates relative conditionalprobabilities to minimum-error classification boundaries. This leads us toformulate an optimization problem that: (i) embeds the data in a parameterized,curved space; (ii) classifies samples based on their position relative to acoordinate axis; and (iii) subsequently optimizes the space by minimizing theempirical classification error of pure training data, for which the classes areknown. Interestingly, the solution to this problem requires use of ahomotopy-type method to stabilize the optimization. We then extend the analysisto the case of impure training data, for which the classes are unknown. We findthat two impure datasets suffice for both prevalence estimation andclassification, provided they satisfy a linear independence property. Lastly,we discuss how our analysis unifies discriminative and generative learningtechniques in a common framework based on ideas from set and measure theory.Throughout, we validate our methods in the context of synthetic data and aresearch-use SARS-CoV-2 enzyme-linked immunosorbent (ELISA) assay.

Minimal Assumptions for Optimal Serology Classification: Theory and Implications for Multidimensional Settings and Impure Training Data