树张量网络的逼近理论：张量化单变量函数——第二部分
Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part II
AI总结：
该研究作为树张量网络逼近理论的第二部分，通过将经典逼近工具编码为可控复杂度张量网络，推导了直接不等式，证明Besov空间连续嵌入其逼近空间且Besov函数可近最优逼近，同时指出无深度限制时逼近类函数未必有Besov光滑性。
摘要：
我们研究了经典光滑类函数的张量网络（TNs）逼近问题。所采用的逼近工具结合了$L^p([0,1))$中函数的张量化（可将单变量函数等价为多变量函数或张量）与树张量网络（张量列格式），以利用多变量函数的低秩结构。该工具可被解释为前馈神经网络：前几层实现张量化，对应一种特定的特征提取步骤，随后是具有稀疏架构的和积网络。在本研究的第一部分中，我们提出了若干与张量网络不同复杂度度量相关的逼近类，并研究了其性质。在本研究（第二部分）中，我们展示了多项式、样条（固定或自由节点）等经典逼近工具如何被编码为复杂度可控的张量网络。我们借此推导了张量网络逼近空间的直接（Jackson）不等式，进而证明Besov空间连续嵌入这些逼近空间。换言之，我们证明任意Besov函数都能以最优或接近最优的速率被逼近。我们还表明，除非限制张量网络的深度，否则逼近类中的任意函数不一定具有Besov光滑性。
英文摘要：
We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.
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