We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.

翻译：本文研究以$L_p$范数度量误差时，Sobolev空间与Besov空间的流形$n$-宽度确定问题。流形宽度控制了这些空间通过一般非线性参数化方法逼近的效率，其中参数选择映射与参数化映射必须满足连续性条件。现有上下界仅在Sobolev或Besov光滑指数$q$满足$q\leq p$或$1 \leq p \leq 2$时相匹配。我们填补了这一空白，对紧嵌入成立的所有$1 \leq p,q \leq \infty$范围给出了尖锐下界。分析的关键在于确定当$p\leq q$时，有限维$\ell^M_q$球在$\ell_p$范数下流形宽度的精确值。尽管该结果并非全新发现，我们提供了新的证明方法，并将其应用于Sobolev与Besov空间流形宽度的下界估计。研究结果表明，通常用于流形宽度下界估计的Bernstein宽度，在许多情况下具有比流形宽度更快的渐近衰减速率。