  This work is motivated by multimodality breast cancer imaging data, which is
quite challenging in that the signals of discrete tumor-associated
microvesicles (TMVs) are randomly distributed with heterogeneous patterns. This
imposes a significant challenge for conventional imaging regression and
dimension reduction models assuming a homogeneous feature structure. We develop
an innovative multilayer tensor learning method to incorporate heterogeneity to
a higher-order tensor decomposition and predict disease status effectively
through utilizing subject-wise imaging features and multimodality information.
Specifically, we construct a multilayer decomposition which leverages an
individualized imaging layer in addition to a modality-specific tensor
structure. One major advantage of our approach is that we are able to
efficiently capture the heterogeneous spatial features of signals that are not
characterized by a population structure as well as integrating multimodality
information simultaneously. To achieve scalable computing, we develop a new
bi-level block improvement algorithm. In theory, we investigate both the
algorithm convergence property, tensor signal recovery error bound and
asymptotic consistency for prediction model estimation. We also apply the
proposed method for simulated and human breast cancer imaging data. Numerical
results demonstrate that the proposed method outperforms other existing
competing methods.
