In this work we develop new model reduction approaches to predict the solutions of time-dependent parametrized problems describing crowd motion in the presence of obstacles. The problem of interest is described by a discrete contact model (DCM), which is formulated as a constrained least-squares optimization statement: the velocity field is seeked as the projection of the spontaneous velocities of each particle into a feasibility set (which prevents the overlap between particles and between particles and obstacles). The parametric variations in the problem are associated to the geometric configuration of the system and to the initial positions of the particles: they have a dramatic impact in the solution, both in terms of particles positions and contact forces. We investigate new developments of the reduced-basis method and supervised machine-learning techniques to effectively find, in a decorrelated manner, primal and dual reduced spaces. We perform a non-linear reconstruction from the first coordinates of a linear reduced basis approximation, in order to achieve a better performance than the linear reconstruction. To assess the validity of the method, the nonlinear compressive strategy is then compared to more standard linear and nonlinear approximations.