Single-cell sequencing technology has been widely used in studies of many biological 36 systems, including but not limited to embryonic development (Xue et netNMF-sc (Elyanow et al. 2020) , each of which seeks to improve recovery of the 61 expression matrix for single-cell data. However, for datasets with high dropout rates-62 which therefore have very sparse expression matrices-it is still a challenge to 63 wherein a higher ARI value indicates that the clustering result is relatively closer to the 108 "true" cell types. Using the expression matrix imputed by WEDGE, we can clearly 109 distinguish these cell types. The ARI value of the cell clusters from the WEDGE 110 imputed matrix is 0.99, higher than those from the other three imputation methods. To examine the performance of WEDGE on real scRNA-seq data, we applied it to 144 Zeisel's dataset (Zeisel et al. 2015 ) on mouse brain scRNA-seq. We first constructed 145 all the genes detected in more than 40% of cells, and then generated an "observed" 147 matrix with high sparsity by randomly setting a large proportion of the non-zero 148 elements to zeros (dropout rate=0.85). From the heatmaps of gene expression 149 matrices ( Fig. 2A) , we can see that WEDGE recovered the expression of the DE genes, 150 especially those differentially expressed between interneurons and S1 pyramidal cells. CMD indicates that the imputed data is closer to the reference data (Fig. 2C ). For the 163 matrix generated by WEDGE, the cell-to-cell CMD is 0.03 and the gene-to-gene CMD 164 is 0.12, which are each tied for the lowest of all the tested methods. These comparisons 165 together highlight that our WEDGE approach can recover both the cell-cell and gene-166 gene correlations from sparse single-cell RNA-seq datasets.
In the tSNE map of cells, WEDGE can clearly distinguish interneurons, S1 pyramidal 168 neurons, and CA1 pyramidal neurons, and the ARI value of 0.56 for the clustering 169 result calculated from its imputed matrix is the highest among all tested methods ( ALRA, and ENHANCE enhanced the expression of Cr2 and Fcer2a in some cells, but 253 these methods also amplified batch effects, and clustering based on the imputation 254 data from these methods did not clearly distinguish splenic B cells into FO and MZ 255 subpopulations. SAVER-X did not classify the FO and MZ subpopulations based on 256 differential expression trends for Cr2 or Fcer2a. In addition, VIPER and SCRABBLE 257 were unable to obtain imputation results from this dataset within 100 hours on the 258 computer with 72 CPU-cores (2.2GHz) and 1TB memory, and netNMF-sc did not 259 complete because of memory errors. 1.00; Fig. 4D ). Notably, the DE genes of 277 cluster 9 generated from the WEDGE imputed data cover 99% of the DE genes in the 278 raw data (Fig. 4C) . The WEDGE imputed data also increased the expression bias of to complete the imputation process, which was close to the MAGIC method. (Fig.  301 5A&B). To further assess the computer resources that WEDGE spends on datasets of 302 various sizes, we applied it to impute datasets comprising different numbers of cells 303 (5000~1000000) but a fixed number of genes (2000), which were sampled from the 304 mouse brain atlas project (see Methods). The runtime of WEDGE increased linearly 305 with the number of cells, and its speed was close to DCA and MAGIC (Fig. 5C) . For 306 the dataset containing 1 million cells and 2000 genes, WEDGE finished the imputation 307 of missing values in 12 minutes. Notably, WEDGE offers a visual interactive interface, 308 making it convenient for researchers to use. We have uploaded WEDGE and the 309 datasets used in this study to GitHub (https://github.com/QuKunLab/WEDGE). 310 Here, we present an approach, WEDGE, to impute missing gene expression 320 information in single-cell sequencing datasets that is based on the combination of low-321 rank matrix decomposition and biased weight parameters for the zero and non-zero 322 elements in the expression matrix. We demonstrate that the usage of WEDGE 323 significantly improves the clustering accuracy of many scRNA-seq datasets, amplifies 324 the contribution of differential genes to identifying cell types, and helps distinguish 325 more cell subpopulations from low-quality data. 326 recovery performance is insensitive to for datasets with dropout rates less than 0.6 328 (Supplemental Fig. S11 ; see Methods). For the datasets with dropout rates greater 329 than 0.6, λ values between 0.1~0.15 can produce the best recovery results. We thus 330 set = 0.15 for all datasets presented in this paper. The imputation contribution of the 331 zero elements decreases with the increase of matrix sparsity, but it cannot be ignored, 332 which implies that some zero elements may be related to the low expression of certain 333 genes, rather than simply reflecting experimental noise. 334
There are still challenges for the informative imputation of scRNA-seq datasets, such 335 as how to recover the heterogeneity between cell types instead of experimental 336 batches, how to discover cell subtypes with very few cells from the imputed data, and 337 how to use limited computer resources to process large datasets containing millions of 338 cells. Moreover, it necessary to assess whether current imputation methods are 339 applicable to datasets obtained using diverse bioanalytical methods beyond standard 340 RNA-seq (e.g., single-cell ATAC-seq and profiling methods for various epigenomic 341 modifications). Users can tune ∈ [0, 1] to balance the contributions of the two terms of the objective 362 functions. In order to study the influence of on the recovery performance, we down-363 sampled the reference data of the Zeisel and Baron datasets and adopted different 364 dropout rates. By computing the correlation matrix distances (CMDs) ( 
where ̃ is the combination of + and 0 according to the original order of their 380 elements in , and is the ith row of . In this case, optimizing is equivalent 381 to solving non-negative least-squares problems (2) in parallel (Lawson and Hanson 382 1995). After was obtained, we fixed it and solved using similar algorithm as 383 described above. Step2: from a given , solve in parallel with a non-negative least-square method.
Step3: from the obtained in step 2, calculate a new .
Step4: iteratively return back to step 2 and 3 until the relative difference in the object function between two adjacent loops is less than 1×10 -5 or the maximum specified number of iterations is reached.
CMD is usually used to determine the difference between two correlation matrices. It With the fixed number of genes, we sampled 1000, 5000, 10000, 100000, 500000, and 501 1000000 cells from the raw dataset to simulate experiments of different scales. 502
(1) Settings for dimension reduction: For the simulated dataset, Baron's dataset, and 505
Zeisel's datasets, we used the first 20 principal components to perform tSNE analysis. 