  In many categorical response regression applications, the response categories
admit a multiresolution structure. That is, subsets of the response categories
may naturally be combined into coarser response categories. In such
applications, practitioners are often interested in estimating the resolution
at which a predictor affects the response category probabilities. In this
article, we propose a method for fitting the multinomial logistic regression
model in high dimensions that addresses this problem in a unified and
data-driven way. In particular, our method allows practitioners to identify
which predictors distinguish between coarse categories but not fine categories,
which predictors distinguish between fine categories, and which predictors are
irrelevant. For model fitting, we propose a scalable algorithm that can be
applied when the coarse categories are defined by either overlapping or
nonoverlapping sets of fine categories. Statistical properties of our method
reveal that it can take advantage of this multiresolution structure in a way
existing estimators cannot. We use our method to model cell type probabilities
as a function of a cell's gene expression profile (i.e., cell type annotation).
Our fitted model provides novel biological insights which may be useful for
future automated and manual cell type annotation methodology.
