  Solving linear discrete ill-posed problems for third order tensor equations
based on a tensor t-product has attracted much attention. But when the data
tensor is produced continuously, current algorithms are not time-saving. Here,
we propose an incremental tensor regularized least squares (t-IRLS) algorithm
with the t-product that incrementally computes the solution to the tensor
regularized least squares (t-RLS) problem with multiple lateral slices on the
right-hand side. More specifically, we update its solution by solving a t-RLS
problem with a single lateral slice on the right-hand side whenever a new
horizontal sample arrives, instead of solving the t-RLS problem from scratch.
The t-IRLS algorithm is well suited for large data sets and real time
operation. Numerical examples are presented to demonstrate the efficiency of
our algorithm.
