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.

翻译：张量链（TT）分解为压缩网格结构数据（如规则笛卡尔网格上的采样函数值）提供了一个强大的框架。这种高效压缩反过来又支持了高维计算的高效执行。精确的TT表示仅适用于简单的解析函数。此外，全局多项式或傅里叶展开通常会导致TT秩随基函数项数成比例增长。现有最先进方法往往计算成本过高或无法恢复潜在的低秩结构。本文提出一种低秩TT插值框架：给定一个描述粗网格上离散（标量、向量或张量值）函数的TT表示（包含$n$个核心），该框架能构建同一函数的细尺度版本，其TT表示包含$n+m$个核心，且最后$m$个核心保持恒定秩。本方法保证了与核心总数无关的$\ell^{2}$范数误差界，在固定精度下实现指数级压缩，并具有关于网格点数的对数级复杂度。我们通过数值实验验证了其性能，包括一维、二维和三维应用：二维与三维翼型掩码嵌入、图像超分辨率、以及三维合成湍流等合成噪声场。特别地，我们以对数级复杂度和内存需求直接在TT格式中生成分形噪声场。这项工作为开发具有复杂几何结构的可扩展原生TT求解器及多尺度生成模型开辟了新路径，对科学模拟、成像和实时图形等领域具有重要影响。