  This paper introduces a randomized variation of the alternating least squares
(ALS) algorithm for rank reduction of canonical tensor formats. The aim is to
address the potential numerical ill-conditioning of least squares matrices at
each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates
large condition numbers via projections onto random tensors, a technique
inspired by well-established randomized projection methods for solving
overdetermined least squares problems in a matrix setting. A probabilistic
bound on the condition numbers of the randomized ALS matrices is provided,
demonstrating reductions relative to their standard counterparts. Additionally,
results are provided that guarantee comparable accuracy of the randomized ALS
solution at each iteration. The performance of the randomized algorithm is
studied with three examples, including manufactured tensors and an elliptic PDE
with random inputs. In particular, for the latter, tests illustrate not only
improvements in condition numbers, but also improved accuracy of the iterative
solver for the PDE solution represented in a canonical tensor format.
