  We present new convergence estimates of generalized empirical interpolation
methods in terms of the entropy numbers of the parametrized function class. Our
analysis is transparent and leads to sharper convergence rates than the
classical analysis via the Kolmogorov n-width. In addition, we also derive
novel entropy-based convergence estimates of the Chebyshev greedy algorithm for
sparse n-term nonlinear approximation of a target function. This also improves
classical convergence analysis when corresponding entropy numbers decay fast
enough.
