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Abstract MotivationSingle-cell Hi-C (scHi-C) technologies have significantly advanced our understanding of the 3D genome organization. However, scHi-C data are often sparse and noisy, leading to substantial computational challenges in downstream analyses. ResultsIn this study, we introduce SHICEDO, a novel deep-learning model specifically designed to enhance scHi-C contact matrices by imputing missing or sparsely captured chromatin contacts through a generative adversarial framework. SHICEDO leverages the unique structural characteristics of scHi-C matrices to derive customized features that enable effective data enhancement. Additionally, the model incorporates a channel-wise attention mechanism to mitigate the over-smoothing issue commonly associated with scHi-C enhancement methods. Through simulations and real-data applications, we demonstrate that SHICEDO outperforms the state-of-the-art methods, achieving superior quantitative and qualitative results. Moreover, SHICEDO enhances key structural features in scHi-C data, thus enabling more precise delineation of chromatin structures such as A/B compartments, TAD-like domains, and chromatin loops. Availability and implementationSHICEDO is publicly available at https://github.com/wmalab/SHICEDO. 

Abstract We show that for every integer and large , every properly edge‐colored graph on vertices with at least edges contains a rainbow subdivision of . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.