Local Interpolation via Low-Rank Tensor Trains

Tensor Train (TT) decompositions provide a powerful framework to compress grid-structured data, such as sampled function values, on regular Cartesian grids. Such high compression, in turn, enables efficient high-dimensional computations. Exact TT representations are only available for simple analytic functions. Furthermore, global polynomial or Fourier expansions typically yield TT-ranks that grow proportionally with the number of basis terms. State-of-the-art methods are often prohibitively expensive or fail to recover the underlying low-rank structure. We propose a low-rank TT interpolation framework that, given a TT describing a discrete (scalar-, vector-, or tensor-valued) function on a coarse regular grid with $n$ cores, constructs a finer-scale version of the same function represented by a TT with $n+m$ cores, where the last $m$ cores maintain constant rank. Our method guarantees a $\ell^{2}$-norm error bound independent of the total number of cores, achieves exponential compression at fixed accuracy, and admits logarithmic complexity with respect of the number of grid points. We validate its performance through numerical experiments, including 1D, 2D, and 3D applications such as: 2D and 3D airfoil mask embeddings, image super-resolution, and synthetic noise fields such as 3D synthetic turbulence. In particular, we generate fractal noise fields directly in TT format with logarithmic complexity and memory. This work opens a path to scalable TT-native solvers with complex geometries and multiscale generative models, with implications from scientific simulation to imaging and real-time graphics.