HaTT: Hadamard avoiding TT recompression

Abstract: The Hadamard product of tensor train (TT) tensors is one of the most fundamental nonlinear operations in scientific computing and data analysis. Due to its tendency to significantly increase TT ranks, the Hadamard product presents a major computational challenge in TT tensor-based algorithms. Therefore, it is essential to develop recompression algorithms that mitigate the effects of this rank increase. Existing recompression algorithms require an explicit representation of the Hadamard product, resulting in high computational and storage complexity. In this work, we propose the Hadamard avoiding TT recompression (HaTT) algorithm. Leveraging the structure of the Hadamard product in TT tensors and its Hadamard product-free property, the overall complexity of the HaTT algorithm is significantly lower than that of existing TT recompression algorithms. This is validated through complexity analysis and several numerical experiments.