量化张量列的多尺度插值构造
Multiscale interpolative construction of quantized tensor trains
AI总结：
本文从多尺度多项式插值视角推进了量化张量列（QTT）近似理论，阐明了QTT秩随深度衰减及平滑性对其的影响，并提出了基于多分辨率网格构造QTT的高效算法。
摘要：
量化张量列（QTTs）最近作为一种用于连续函数数值离散化的框架出现，在数值分析中具有广泛应用潜力。然而，QTT近似理论尚未被完全理解。在本工作中，我们从多尺度多项式插值的角度推进了该理论。这一视角阐明了QTT秩为何随深度增加而衰减，定量地根据目标函数的平滑性控制QTT秩，并解释了为何某些具有尖锐特征和较差定量平滑性的函数仍能被QTTs很好地近似。该视角还启发了基于多分辨率网格上函数评估来构造QTTs的全新实用且高效的算法。
英文摘要：
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.
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