Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis. However, the theory of QTT approximation is not fully understood. In this work, we advance this theory from the point of view of multiscale polynomial interpolation. This perspective clarifies why QTT ranks decay with increasing depth, quantitatively controls QTT rank in terms of smoothness of the target function, and explains why certain functions with sharp features and poor quantitative smoothness can still be well approximated by QTTs. The perspective also motivates new practical and efficient algorithms for the construction of QTTs from function evaluations on multiresolution grids.

翻译：量化张量列（QTT）近年来作为一种连续函数数值离散化框架出现，在数值分析中具有广泛的应用前景。然而，QTT逼近理论尚未被完全理解。本文从多尺度多项式插值的角度推进了这一理论。该视角阐明了为何QTT秩随着深度增加而衰减，定量地根据目标函数的光滑性控制QTT秩，并解释了为何某些具有尖锐特征且定量光滑性较差的函数仍能被QTT良好逼近。该视角还催生了基于多分辨率网格函数评估构建QTT的新实用高效算法。