Linear Optimal Low Rank Projection for High-Dimensional Multi-Class Data

Classifying samples into categories becomes intractable when a single sample can have millions to billions of features, such as in genetics or imaging data. Principal Components Analysis (PCA) is widely used to identify a low-dimensional representation of such features for further analysis. However, PCA ignores class labels, such as whether or not a subject has cancer, thereby discarding information that could substantially improve downstream classification performance. We describe an approach, "Linear Optimal Low-rank" projection (LOL), which extends PCA by incorporating the class labels in a fashion that is advantageous over existing supervised dimensionality reduction techniques. We prove, and substantiate with synthetic experiments, that LOL leads to a better representation of the data for subsequent classification than other linear approaches, while adding negligible computational cost. We then demonstrate that LOL substantially outperforms PCA in differentiating cancer patients from healthy controls using genetic data, and in differentiating gender using magnetic resonance imaging data with $>$500 million features and 400 gigabytes of data. LOL therefore allows the solution of previous intractable problems, yet requires only a few minutes to run on a desktop computer.