Our presentation takes place within the context of model reduction for parametric partial differential equations (PPDE). When the solution of the PPDE has to be evaluated for many different values the parameters, the computational effort may become prohibitive. To circumvent this issue, model reduction intends to simplify the resolution of the PPDE by (typically) constraining the solutions to belong to some low-dimensional subspace. Many techniques have been proposed in the literature to identify such subspaces: Taylor or Hermite expansions, proper orthogonal decomposition (POD), balanced truncation, reduced basis techniques, etc. All the methods mentioned above presuppose some refined knowledge of the solution manifold M. Unfortunately, in practice a refined knowledge of M may not always be available. The question addressed in our work is therefore as follows: can we still build a "good" approximation subspace if M is imperfectly known but some partial measurements of the latter are available? We propose practical procedures to deal with this problem and provide theoretical results to analyze their behavior.