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：我们研究在$L_p$-范数误差度量下Sobolev空间与Besov空间流形$n$-宽度的确定问题。流形宽度刻画了这些空间通过一般非线性参数化方法（需满足参数选择与参数化映射的连续性约束）的逼近效率。现有上下界仅在Sobolev或Besov光滑指数$q$满足$q\leq p$或$1 \leq p \leq 2$时匹配。本文填补了这一间隙，针对所有满足紧嵌入条件的$1 \leq p,q \leq \infty$参数范围，获得了锐利下界。分析的关键在于确定当$p\leq q$时有限维$\ell^M_q$-球在$\ell_p$-范数下的流形宽度精确值。尽管该结果并非原创，但我们提供了新的证明方法，并将其应用于Sobolev空间与Besov空间流形宽度的下界估计。研究结果表明，通常用于建立流形宽度下界的Bernstein宽度在许多情形下渐近衰减速度更快于流形宽度。