  We study high-dimensional nonlinear approximation of functions in
H\"older-Nikol'skii spaces $H^\alpha_\infty(\mathbb{I}^d)$ on the unit cube
$\mathbb{I}^d:=[0,1]^d$ having mixed smoothness, by parametric manifolds. The
approximation error is measured in the $L_\infty$-norm. In this context, we
explicitly constructed methods of nonlinear approximation, and give
dimension-dependent estimates of the approximation error explicitly in
dimension $d$ and number $N$ measuring computation complexity of the parametric
manifold of approximants. For $d=2$, we derived a novel right asymptotic order
of noncontinuous manifold $N$-widths of the unit ball of
$H^\alpha_\infty(\mathbb{I}^2)$ in the space $L_\infty(\mathbb{I}^2)$. In
constructing approximation methods, the function decomposition by the tensor
product Faber series and special representations of its truncations on sparse
grids play a central role.
