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）近年来作为连续函数数值离散化框架出现，在数值分析领域具有广泛的应用潜力。然而，QTT逼近理论尚未被完全理解。本文从多尺度多项式插值的视角推进该理论发展：该视角阐明了QTT秩随深度增加而衰减的原因，定量描述了QTT秩与目标函数光滑性的关系，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。该视角还启发了一种新型实用高效的算法，用于从多分辨率网格上的函数评估中构建QTT。