In this work, we propose an application to dipole-dipole interactions of a data compression method, called quantics tensor train cross interpolation [1,2,3], that has been developed over the last decade in computational mathematics. It is inspired by the matrix product representation of quantum states in 1D chains, and leverages the optimisation algorithms in linear algebra for machine learning. I introduce the technique in three steps; unfolding of large arrays into a train of smaller sized tensors, a clever rearranging of tensor indices that enable logarithmic computational complexity in the system size in many cases of physical interest, and the cross interpolation algorithm that makes the data compression suitable for iterative manipulations. I will present the tensor train representation of the kernel of dipole-dipole interactions which show a dramatic reduction of the data size (Fig. 1). Preliminary results of some model calculations that illustrate the scope of potential applications will also be discussed.