The approximation of a high-dimensional vector by a small combination of column vectors selected from a fixed matrix has been actively debated in several different disciplines. In this paper, a sampling approach based on the Monte Carlo method is presented as an efficient solver for such problems. Especially, the use of simulated annealing (SA), a metaheuristic optimization algorithm, for determining degrees of freedom (the number of used columns) by cross validation is focused on and tested. Test on a synthetic model indicates that our SA-based approach can find a nearly optimal solution for the approximation problem and, when combined with the CV framework, it can optimize the generalization ability. Its utility is also confirmed by application to a real-world supernova data set.