  We present function preserving projections (FPP), a scalable linear
projection technique for discovering interpretable relationships in
high-dimensional data. Conventional dimension reduction methods aim to
maximally preserve the global and/or local geometric structure of a dataset.
However, in practice one is often more interested in determining how one or
multiple user-selected response function(s) can be explained by the data. To
intuitively connect the responses to the data, FPP constructs 2D linear
embeddings optimized to reveal interpretable yet potentially non-linear
patterns of the response functions. More specifically, FPP is designed to (i)
produce human-interpretable embeddings; (ii) capture non-linear relationships;
(iii) allow the simultaneous use of multiple response functions; and (iv) scale
to millions of samples. Using FPP on real-world datasets, one can obtain
fundamentally new insights about high-dimensional relationships in large-scale
data that could not be achieved using existing dimension reduction methods.
