In the present paper we develop two algorithms, Subset Based Training (SBT) and Subset Based Training and Pruning (SBTP), using the fact that the Jacobian matrices in sigmoid network training problems are usually rank deficient. The weight vectors are divided into two parts during training, according to the Jacobian rank sizes. Both SBT and SBTP are trust region methods. Comparing to the standard Levenberg-Marquardt (LM) method, these two algorithms can achieve similar convergence properties as the LM but with less memory requirements. Furthermore the SBTP combines training and pruning of a network into one comprehensive procedure. Some convergence properties of the two algorithms are given to qualitatively evaluate the performance of the algorithms.