  We consider the problem of tensor estimation from noisy observations with
possibly missing entries. A nonparametric approach to tensor completion is
developed based on a new model which we coin as sign representable tensors. The
model represents the signal tensor of interest using a series of structured
sign tensors. Unlike earlier methods, the sign series representation
effectively addresses both low- and high-rank signals, while encompassing many
existing tensor models -- including CP models, Tucker models, single index
models, several hypergraphon models -- as special cases. We show that the sign
tensor series is theoretically characterized, and computationally estimable,
via classification tasks with carefully-specified weights. Excess risk bounds,
estimation error rates, and sample complexities are established. We demonstrate
the outperformance of our approach over previous methods on two datasets, one
on human brain connectivity networks and the other on topic data mining.
