  Tensor completion aims at imputing missing entries from a partially observed
tensor. Existing tensor completion methods often assume either multi-linear or
nonlinear relationships between latent components.
  However, real-world tensors have much more complex patterns where both
multi-linear and nonlinear relationships may coexist. In such cases, the
existing methods are insufficient to describe the data structure. This paper
proposes a Joint mUlti-linear and nonLinear IdentificAtion (JULIA) framework
for large-scale tensor completion. JULIA unifies the multi-linear and nonlinear
tensor completion models with several advantages over the existing methods: 1)
Flexible model selection, i.e., it fits a tensor by assigning its values as a
combination of multi-linear and nonlinear components; 2) Compatible with
existing nonlinear tensor completion methods; 3) Efficient training based on a
well-designed alternating optimization approach. Experiments on six real
large-scale tensors demonstrate that JULIA outperforms many existing tensor
completion algorithms. Furthermore, JULIA can improve the performance of a
class of nonlinear tensor completion methods. The results show that in some
large-scale tensor completion scenarios, baseline methods with JULIA are able
to obtain up to 55% lower root mean-squared-error and save 67% computational
complexity.
