Sharp lower bounds on the manifold widths of Sobolev and Besov spaces

We study the manifold n-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths measure how efficiently these spaces can be approximated by continuous non-linear parametric methods. Existing upper and lower bounds only match when the smoothness index q satisfies $q\le p$ or $1\le p\le 2$. We close this gap, obtaining sharp bounds for all $1\le p,q\le \infty$ for which a compact embedding holds. In the process, we determine the exact value of the manifold widths of finite dimensional ${\ell }_q^M$-balls in the ${\ell }_p$-norm when $p\le q$. 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.