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We consider the problem of estimating the geographic locations of nodes in a wireless sensor network where most sensors are without an effective self-positioning functionality. We propose LSVM-a novel solution with the following merits. First, LSVM localizes the network based on mere connectivity information (that is, hop counts only) and therefore is simple and does not require specialized ranging hardware or assisting mobile devices as in most existing techniques. Second, LSVM is based on Support Vector Machine (SVM) learning. Although SVM is a classification method, we show its applicability to the localization problem and prove that the localization error can be upper bounded by any small threshold given an appropriate training data size. Third, LSVM addresses the border and coverage-hole problems effectively. Last but not least, LSVM offers fast localization in a distributed manner with efficient use of processing and communication resources. We also propose a modified version of mass-spring optimization to further improve the location estimation in LSVM. The promising performance of LSVM is exhibited by our simulation study.