Low-rank nonnegative tensor approximation via alternating projections and sketching

We show how to construct nonnegative low-rank approximations of nonnegative tensors in Tucker and tensor train formats. We use alternating projections between the nonnegative orthant and the set of low-rank tensors, using STHOSVD and TTSVD algorithms, respectively, and further accelerate the alternating projections using randomized sketching. The numerical experiments on both synthetic data and hyperspectral images show the decay of the negative elements and that the error of the resulting approximation is close to the initial error obtained with STHOSVD and TTSVD. The proposed method for the Tucker case is superior to the previous ones in terms of computational complexity and decay of negative elements. The tensor train case, to the best of our knowledge, has not been studied before.