  Learning of matrix-valued data has recently surged in a range of scientific
and business applications. Trace regression is a widely used method to model
effects of matrix predictors and has shown great success in matrix learning.
However, nearly all existing trace regression solutions rely on two
assumptions: (i) a known functional form of the conditional mean, and (ii) a
global low-rank structure in the entire range of the regression function, both
of which may be violated in practice. In this article, we relax these
assumptions by developing a general framework for nonparametric trace
regression models via structured sign series representations of high
dimensional functions. The new model embraces both linear and nonlinear trace
effects, and enjoys rank invariance to order-preserving transformations of the
response. In the context of matrix completion, our framework leads to a
substantially richer model based on what we coin as the "sign rank" of a
matrix. We show that the sign series can be statistically characterized by
weighted classification tasks. Based on this connection, we propose a learning
reduction approach to learn the regression model via a series of classifiers,
and develop a parallelable computation algorithm to implement sign series
aggregations. We establish the excess risk bounds, estimation error rates, and
sample complexities. Our proposal provides a broad nonparametric paradigm to
many important matrix learning problems, including matrix regression, matrix
completion, multi-task learning, and compressed sensing. We demonstrate the
advantages of our method through simulations and two applications, one on brain
connectivity study and the other on high-rank image completion.
