Variations in DNA copy number carry important information on genome evolution and regulation of DNA replication in cancer cells. The rapid development of single-cell sequencing technology enables exploration of gene-expression heterogeneity among single cells, providing important information on cell evolution. Evolutionary relationships in accumulated sequence data can be visualized by adjacent positioning of similar cells so that similar copy-number profiles are shown by block patterns. However, single-cell DNA sequencing data usually have low amount of starting genome, which requires an extra step of amplification to accumulate sufficient samples, introducing noise and making regular pattern-finding challenging. In this paper, we will propose to tackle this issue of recovering the hidden blocks within single-cell DNA-sequencing data through continuous sample permutations such that similar samples are positioned adjacently. The permutation is guided by the total variational norm of the recovered copy number profiles, and is continued until the total variational norm is minimized when similar samples are stacked together to reveal block patterns. An efficient numerical scheme for finding this permutation is designed, tailored from the alternating direction method of multipliers. Application of this method to both simulated and real data demonstrates its ability to recover the hidden structures of single-cell DNA sequences.
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