  To solve key biomedical problems, experimentalists now routinely measure
millions or billions of features (dimensions) per sample, with the hope that
data science techniques will be able to build accurate data-driven inferences.
Because sample sizes are typically orders of magnitude smaller than the
dimensionality of these data, valid inferences require finding a
low-dimensional representation that preserves the discriminating information
(e.g., whether the individual suffers from a particular disease). There is a
lack of interpretable supervised dimensionality reduction methods that scale to
millions of dimensions with strong statistical theoretical guarantees.We
introduce an approach, XOX, to extending principal components analysis by
incorporating class-conditional moment estimates into the low-dimensional
projection. The simplest ver-sion, "Linear Optimal Low-rank" projection (LOL),
incorporates the class-conditional means. We prove, and substantiate with both
synthetic and real data benchmarks, that LOL and its generalizations in the XOX
framework lead to improved data representations for subsequent classification,
while maintaining computational efficiency and scalability. Using multiple
brain imaging datasets consisting of >150 million features, and several
genomics datasets with>500,000 features, LOL outperforms other scalable linear
dimensionality reduction techniques in terms of accuracy, while only requiring
a few minutes on a standard desktop computer.
