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Local linearization techniques are an important class of nonparametric system identification. Identifying local linearizations in practice involves solving a linear regression problem that is ill-posed. The problem can be ill-posed either if the dynamics of the system lie on a manifold of lower dimension than the ambient space or if there are not enough measurements of all the modes of the dynamics of the system. We describe a set of linear regression estimators that can handle data lying on a lower-dimension manifold. These estimators differ from previous estimators, because these estimators are able to improve estimator performance by exploiting the sparsity of the system - the existence of direct interconnections between only some of the states - and can work in the ldquolarge p, small nrdquo setting in which the number of states is comparable to the number of data points. We describe our system identification procedure, which consists of a pre smoothing step and a regression step, and then we apply this procedure to data taken from a quadrotor helicopter. We use this data set to compare our procedure with existing procedures.