  Positive-valued signal data is common in many biological and medical
applications, where the data are often generated from imaging techniques such
as mass spectrometry. In such a setting, the relative intensities of the raw
features are often the scientifically meaningful quantities, so it is of
interest to identify relevant features that take the form of log-ratios of the
raw inputs. When including the log-ratios of all pairs of predictors, the
dimensionality of this predictor space becomes large, so computationally
efficient statistical procedures are required. We introduce an embedding of the
log-ratio parameter space into a space of much lower dimension and develop
efficient penalized fitting procedure using this more tractable representation.
This procedure serves as the foundation for a two-step fitting procedure that
combines a convex filtering step with a second non-convex pruning step to yield
highly sparse solutions. On a cancer proteomics data set we find that these
methods fit highly sparse models with log-ratio features of known biological
relevance while greatly improving upon the predictive accuracy of less
interpretable methods.
