  We show that for complex nonlinear systems, model reduction and compressive
sensing strategies can be combined to great advantage for classifying,
projecting, and reconstructing the relevant low-dimensional dynamics.
$\ell_2$-based dimensionality reduction methods such as the proper orthogonal
decomposition are used to construct separate modal libraries and Galerkin
models based on data from a number of bifurcation regimes. These libraries are
then concatenated into an over-complete library, and $\ell_1$ sparse
representation in this library from a few noisy measurements results in correct
identification of the bifurcation regime. This technique provides an objective
and general framework for classifying the bifurcation parameters, and
therefore, the underlying dynamics and stability. After classifying the
bifurcation regime, it is possible to employ a low-dimensional Galerkin model,
only on modes relevant to that bifurcation value. These methods are
demonstrated on the complex Ginzburg-Landau equation using sparse, noisy
measurements. In particular, three noisy measurements are used to accurately
classify and reconstruct the dynamics associated with six distinct bifurcation
regimes; in contrast, classification based on least-squares fitting ($\ell_2$)
fails consistently.
