Approximation of high-dimensional functions is a problem in many scientific fields that is only feasible if advantageous structural properties, such as sparsity in a given basis, can be exploited. A relevant tool for analysing sparse approximations is Stechkin's lemma. In its standard form, however, this lemma does not allow to explain convergence rates for a wide range of relevant function classes. This work presents a new weighted version of Stechkin's lemma that improves the best $n$-term rates for weighted $\ell^p$-spaces and associated function classes such as Sobolev or Besov spaces. For the class of holomorphic functions, which occur as solutions of common high-dimensional parameter-dependent PDEs, we recover exponential rates that are not directly obtainable with Stechkin's lemma. Since weighted $\ell^p$-summability induces weighted sparsity, compressed sensing algorithms can be used to approximate the associated functions. To break the curse of dimensionality, which these algorithms suffer, we recall that sparse approximations can be encoded efficiently using tensor networks with sparse component tensors. We also demonstrate that weighted $\ell^p$-summability induces low ranks, which motivates a second tensor train format with low ranks and a single weighted sparse core. We present new alternating algorithms for best $n$-term approximation in both formats. To analyse the sample complexity for the new model classes, we derive a novel result of independent interest that allows the transfer of the restricted isometry property from one set to another sufficiently close set. Although they lead up to the analysis of our final model class, our contributions on weighted Stechkin and the restricted isometry property are of independent interest and can be read independently.

翻译：高维函数逼近是众多科学领域中的难题，唯有利用有利的结构特性（如给定基下的稀疏性）才可能实现可行逼近。分析稀疏逼近的重要工具之一是斯捷奇金引理。然而，该引理的标准形式无法解释广泛相关函数类的收敛速率。本文提出了斯捷奇金引理的新型加权版本，该版本改进了加权$\ell^p$空间及关联函数类（如索伯列夫空间或贝索夫空间）的最佳$n$项逼近速率。对于作为常见高维参数依赖偏微分方程解的全纯函数类，我们恢复了指数收敛速率——这是标准斯捷奇金引理无法直接获得的。由于加权$\ell^p$可和性诱导加权稀疏性，压缩感知算法可用于逼近相关函数。为打破此类算法固有的维数灾难，我们指出可通过稀疏分量张量的张量网络高效编码稀疏逼近。同时证明加权$\ell^p$可和性可诱导低秩结构，由此提出第二种含低秩与单个加权稀疏核的张量链格式。针对两种格式的最佳$n$项逼近，我们给出新型交替算法。为分析新模型类的样本复杂度，我们推导出一个具有独立价值的新结论，该结论可将受限等距性质从一个集合传递至足够接近的另一集合。尽管上述成果最终服务于最终模型类的分析，但关于加权斯捷奇金引理和受限等距性质的贡献具有独立学术价值，可单独研读。