Tensor Train Completion of Multi-Way Data Observed Along One Mode

Tensor completion is an extension of matrix completion aimed at recovering a tensor from partial observations. Many problems can be framed as instances of low-rank tensor completion. Typically, the low-rank tensor completion problems are solved by optimization-based algorithms. This work shows how to use only standard linear algebra tools to compute the tensor train decomposition of an incomplete tensor with fibers observed along a single mode. From an application viewpoint, this setting is relevant when it is easier to sample or collect a dataset along a specific mode (e.g., temporal).