Tensorstructured

Tensorstructured refers to a class of computational methods and mathematical objects that exploit the underlying low-rank tensor structure often present in large-scale data and scientific computations. Tensors, which are multi-dimensional arrays, can become prohibitively large in their full representation. Tensorstructured methods aim to represent these tensors in a compressed or factorized form, leveraging their inherent structure to reduce computational complexity and memory requirements.

Key to tensorstructured methods are tensor decomposition techniques, such as Tucker decomposition and tensor train (TT) decomposition. These decompositions represent a high-dimensional tensor as a sum or product of lower-dimensional tensors. This factorization allows for operations like matrix-vector multiplication, tensor addition, and solving linear systems to be performed much more efficiently than with the full tensor.

The applications of tensorstructured methods are widespread, including areas like quantum mechanics, computational fluid dynamics, machine learning, and signal processing. In machine learning, for example, tensor decompositions can be used for dimensionality reduction, recommendation systems, and compressed sensing. The efficiency gains offered by these methods make it feasible to tackle problems that would otherwise be computationally intractable due to the curse of dimensionality. Research in this field focuses on developing new decomposition formats, efficient algorithms for tensor operations, and adapting these techniques to specific application domains.