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）近年来已成为连续函数数值离散化的框架，在数值分析领域具有广泛的应用潜力。然而，QTT逼近理论尚未得到充分理解。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QTT秩随深度增加而衰减的原因，从目标函数光滑性的角度定量控制了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。该视角还启发了基于多分辨率网格函数评估构建QTT的新型实用高效算法。