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We propose a joint channel estimation and signal detection approach for the uplink non-orthogonal multiple access (NOMA) using unsupervised machine learning. We apply a Gaussian mixture model (GMM) to cluster the received signals, and accordingly optimize the decision regions to enhance the symbol error rate (SER) performance. We show that, when the received powers of the users are sufficiently different, the proposed clustering-based approach achieves an SER performance on a par with that of the conventional maximum-likelihood detector (MLD) with full channel state information (CSI). We study the tradeoff between the accuracy of the proposed approach and the blocklength, as the accuracy of the utilized clustering algorithm depends on the number of symbols available at the receiver. We provide a comprehensive performance analysis of the proposed approach and derive a theoretical bound on its SER performance. Our simulation results corroborate the effectiveness of the proposed approach and verify that the calculated theoretical bound can predict the SER performance of the proposed approach well. We further explore the application of the proposed approach to a practical grant-free NOMA scenario, and show that its performance is very close to that of the optimal MLD with full CSI, which usually requires long pilot sequences.