Local Interpolation via Low-Rank Tensor Trains
High-dimensional grid data represented in the tensor train (TT) format often suffer from rank explosion due to global unfolding, hindering efficient interpolation and compression. This work proposes a low-rank TT local interpolation framework that starts from a coarse-grid TT representation and constructs a fine-grid TT with uniformly bounded tail ranks through multiscale local refinement. The method achieves, for the first time, an ℓ² error bound independent of the total number of cores, exponential compression rates at fixed accuracy, and logarithmic computational complexity with respect to the number of grid points. Its efficacy is demonstrated on 1D/2D/3D tasks—including airfoil mask embedding, image super-resolution, and synthetic turbulent noise—and it enables direct generation of fractal noise fields with logarithmic complexity.