Approximating field variables and data vectors from sparse samples is a key\nchallenge in computational science. Widely used methods such as gappy proper\northogonal decomposition and empirical interpolation rely on linear\napproximation spaces, limiting their effectiveness for data representing\ntransport-dominated and wave-like dynamics. To address this limitation, we\nintroduce quadratic manifold sparse regression, which trains quadratic\nmanifolds with a sparse greedy method and computes approximations on the\nmanifold through novel nonlinear projections of sparse samples. The nonlinear\napproximations obtained with quadratic manifold sparse regression achieve\norders of magnitude higher accuracies than linear methods on data describing\ntransport-dominated dynamics in numerical experiments.\n