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Learning and interpolation are two approaches to solve the problem of how to build a reasonable estimate of an unknown function on the basis of a finite number of samples. Such problems arise in various frameworks ranging from partial differential equations through geometric modeling in image synthesis to learning and adaptive control. In this chapter, we present an overview of various existing methods the purpose of which is to estimate functions from samples. 7.1 The learning problem The classical problem that each method has to solve can be stated in the following way: given a number of measures (xn, yn) ∈ R d × R, for n = 1... N, we want to find a function f mapping R d to R such that f(xn) = yn for n = 1... N 7.1.1 What is the best solution? Such a problem can easily be solved in the framework of parametric estimation, where the unknown function is determined by a small number of parameters (as in linear regression). In such a case, the underlying linear system is overdetermined and has usually no solution. A model of measures with a Gaussian noise can be used: yn = f(xn) + ɛn where the ɛn are i.i.d. Gaussian variables of zero mean and of standard deviation σ. Regression consists in finding the parameter combination that maximizes the likelihood (conditional density w.r.t the function f). In the Gaussian case, we maximize