  The tensor cross interpolation (TCI) algorithm is a rank-revealing algorithm
for decomposing low-rank, high-dimensional tensors into tensor trains/matrix
product states (MPS). TCI learns a compact MPS representation of the entire
object from a tiny training data set. Once obtained, the large existing MPS
toolbox provides exponentially fast algorithms for performing a large set of
operations. We discuss several improvements and variants of TCI. In particular,
we show that replacing the cross interpolation by the partially rank-revealing
LU decomposition yields a more stable and more flexible algorithm than the
original algorithm. We also present two open source libraries, xfac in
Python/C++ and TensorCrossInterpolation.jl in Julia, that implement these
improved algorithms, and illustrate them on several applications. These include
sign-problem-free integration in large dimension, the superhigh-resolution
quantics representation of functions, the solution of partial differential
equations, the superfast Fourier transform, the computation of partition
functions, and the construction of matrix product operators.
