Sleep staging aims to gather biological signals during sleep, and categorize them by sleep stages: waking (W), non-REM-1 (N1), non-REM-2 (N2), non-REM-3 (N3), and REM (R). These stages are distributed irregularly, and their number varies with sleep quality. These features adversely affect the performance of automatic sleep staging systems. This paper adopts Siamese neural networks (SNNs) to solve the problem. During the network design, seven distance measurement methods, namely, Euclidean, Manhattan, Jaccard, Cosine, Canberra, Bray-Curtis, and Kullback Leibler divergence (KLD), were compared, revealing that Bray-Curtis (83.52%) and Cosine (84.94%) methods boast the best classification performance. The results of our approach are promising compared to traditional methods.