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We present new convergence estimates of generalized empirical interpolationmethods in terms of the entropy numbers of the parametrized function class. Ouranalysis is transparent and leads to sharper convergence rates than theclassical analysis via the Kolmogorov n-width. In addition, we also derivenovel entropy-based convergence estimates of the Chebyshev greedy algorithm forsparse n-term nonlinear approximation of a target function. This also improvesclassical convergence analysis when corresponding entropy numbers decay fastenough.

A new analysis of empirical interpolation methods and Chebyshev greedy algorithms