scVGAMF: a novel imputation method for scRNA-seq data by integrating linear and non-linear features

Abstract

Single-cell RNA sequencing (scRNA-seq) is crucial for elucidating gene expression dynamics and cellular heterogeneity at the individual cell level, thereby advancing our understanding of transcriptional regulation across distinct cell populations. However, a significant challenge in scRNA-seq data analysis is the prevalence of dropout events, which complicate downstream analyses. Most existing imputation tools either rely solely on linear assumptions or overlook the non-linear regulatory relationships embedded in the data. To address this issue, we propose single-cell variational graph autoencoder and matrix factorization (scVGAMF), a novel imputation method that integrates both linear and non-linear features. Specifically, scVGAMF first identifies highly variable genes and partitions them into groups. Cells are then clustered by applying spectral clustering to the principal component analysis results of the representative groups. Based on the resulting submatrices, along with the gene similarity and cell–cell similarity matrices, scVGAMF employs non-negative matrix factorization to extract underlying linear features while utilizing two variational graph autoencoders to capture non-linear features. A fully connected neural network then integrates these features to predict missing values. Extensive experimental evaluations on simulated dropout datasets and real scRNA-seq data demonstrate that scVGAMF outperforms existing methods in gene expression recovery, cell clustering accuracy, differential gene identification, and pseudo-trajectory analysis. Furthermore, ablation studies confirm that the integration of both linear and non-linear features significantly enhances overall data imputation performance.

Introduction

Single-cell RNA sequencing (scRNA-seq) is an emerging technology that facilitates the analysis of gene expression at the resolution of individual cells [1]. This technology provides unprecedented insights into cellular heterogeneity, transcriptional dynamics, and developmental trajectories. However, scRNA-seq is subject to technical limitations that often lead to undetected gene expression measurements, commonly referred to as “dropout” events. These missing values can distort the true distribution of the data, obscuring crucial gene–gene and cell–cell relationships. As a result, the presence of missing values can significantly impair the accuracy and reliability of downstream analyses, including cell clustering, trajectory inference, and differential expression studies [2–4]. Accurate imputation of these missing values is essential for enhancing the interpretability and validity of scRNA-seq data in biological research.

In recent years, numerous imputation methods have been designed to address “dropout” events [5–26]. One significant category of imputation methods is based on statistical modeling. For example, scImpute [6] applies principal component analysis (PCA) to reduce the dimensionality of the gene expression matrix, then applies spectral clustering to identify cell subpopulations and employs a gamma–Gaussian mixture model to impute missing values. SAVER [7] constructs a Poisson–gamma mixture model and uses Poisson–lasso regression to estimate potential gene expression values. TsImpute [8] distinguishes potential random dropouts from true zeros by estimating the parameters of the zero-inflated negative binomial (ZINB) distribution, and then performs imputation on the preliminarily imputed matrix using inverse distance weighted clustering. Another category of imputation methods is based on data smoothing. MAGIC [9] conducts data diffusion based on Markov affinity matrices, allowing information to be shared between similar cells to infer possible gene expression values in each cell. DrImpute [10] performs multiple imputation by averaging the expression values of similar cells based on different cluster numbers. Low-rank matrix-based methods are also commonly used to capture linear relationships between cells and then reconstruct the gene expression matrix to enable the imputation of missing values. scRMD [11] models robust matrix decomposition and estimates missing values using the alternating direction multiplier method. ALRA [13] uses singular value decomposition to obtain a low-rank approximation of the gene expression matrix that fills in missing values while preserving the zero values in the data. Additionally, significant advances have been made through machine learning, particularly the emergence of graph neural networks (GNNs) [27]. GNN effectively deconvolutes node relationships within graphs by propagating neighbor information. Compared with other types of imputation methods, GNN-based methods aim to derive low-dimensional embeddings of graph topological structures while learning node relationships from a global view of the entire graph’s architecture [28–30]. scGNN [14] integrates three iterative multi-modal autoencoders and aggregates cell-cell relationships with GNNs. scTAG [15] introduced a clustering method that utilizes a topologically adaptive graph convolutional encoder with ZINBLoss for missing value imputation by predicting distributions. GNNImpute [16] employs a graph convolution encoder and a linear decoder to impute the missing values. scVGAE [18] integrates ZINBLoss and graph convolutional network (GCN) to improve imputation while maintaining the integrity of the cellular information.

Despite these advancements that have been made in these imputation methods, several limitations remain. First, many methods alter the original data structure during imputation, potentially distorting biologically significant features. Second, many approaches treat all zero values as technical artifacts, failing to distinguish true biological silences (biological zeros) from dropout-induced false negatives (technical zeros), leading to the risk of over-imputation [21, 31]. Third, many graph-based methods focus heavily on caputring complex patterns in the data, often neglecting linear correlation relationships that could also offer valuable insights. Many methods based on non-negative matrix factorization (NMF) only consider the linear correlation relationships of the data, while overlooking the complex regulatory and coupling relationships between cells and between genes. While linear and non-linear features capture distinct aspects of scRNA-seq data, few methods synergistically combine these complementary perspectives.

To address these limitations, we propose a novel imputation method for scRNA-seq data, named scVGAMF. Initially, scVGAMF employs spectral clustering to distinguish between true and false zeros by analyzing the gene expression rate in different cell clusters. The non-zero values and the distinguished true zeros serve as the training set, while potential dropout values serve as the prediction set. Subsequently, the method combines a variational graph autoencoder (VGAE) for extracting non-linear features and NMF for linear feature extraction. Finally, a fully connected neural network is then applied to combine both non-linear and linear features, generating the final imputed matrix.

To investigate the performance of the scVGAMF, we apply it to eight benchmark datasets and compare the results with ten state-of-the-art imputation methods. Experiments on gene expression recovery, cell clustering, differential gene identification, and pseudo-trajectory analysis show that scVGAMF can accurately impute missing values in scRNA-seq data and improve downstream analysis.

In imputation tasks, integrating non-linear and linear representations allows models to capture multi-scale biological relationships, thereby enhancing prediction accuracy. Based on the VGAMF framework [32], we present scVGAMF, a novel method that integrates VGAEs and NMF to enhance imputation performance. By leveraging the complementary strengths of VGAEs in modeling non-linear dependencies and NMF in capturing linear correlations, scVGAMF achieves more accurate reconstruction of gene expression patterns. Fig. 1 provides an overview of the scVGAMF framework.

Figure 1.

Framework of scVGAMF. Step 1: After log-normalization of the matrix, HVGs are ranked and divided into groups. Step 2: The matrix is divided into submatrices based on the order of HVGs and integrated with the spectral clustering outcomes. Step 3: Non-linear features (captured via VGAE) and linear features (extracted via NMF) are learned from both gene and cell dimensions. Step 4: A fully connected neural network integrates these features to predict missing values.

Unlike conventional methods, scVGAMF integrates interpretable matrix factorization with deep learning, enabling the capture of complex gene–gene interactions while maintaining transparency in the imputation process. The key advantages of scVGAMF include the following:

• Comprehensive handling of linear and non-linear relationships: By integrating both linear features (captured via NMF) and non-linear features (extracted by VGAE), scVGAMF effectively captures diverse biological relationships in scRNA-seq data.

• Interpretable modeling of complex patterns: scVGAMF combines the interpretability of matrix factorization with the pattern-capture capacity of deep learning, producing biologically meaningful and reliable results.

• Enhanced robustness to dropout events: scVGAMF shows improved performance in mitigating the impact of dropout events, which are prevalent in scRNA-seq data.

In summary, scVGAMF represents a significant advance in scRNA-seq imputation by providing a reliable and interpretable framework that effectively addresses dropout events and improves downstream analytical accuracy.

Materials and methods

Data division

For the raw count matrix , logarithmic normalization is applied and the genes are ranked according to the variable value calculated by the variance stabilizing transformation. The genes are then divided into groups (groups of 2000 genes by default). Each gene group is processed separately to obtain . Here, represent the results of each grouping, and denotes the total number of groups. Note that the final group typically contains fewer than 2000 genes. After applying PCA to , we perform spectral clustering. The number of clusters is set to range from 4 to 15 by default. Subsequently, the Silhouette coefficient scores [33] for each clustering result are computed, and the value of that yields the highest scores is selected. At this point , where denotes the set of all cells and represents the clustering results obtained with the selected value. The raw count matrix is then divided according to the clustering results as follows:

(1)

Calculating the similarity matrix

For each submatrix, we compute both the cell similarity matrix and the gene similarity matrix. The cell similarity matrix is constructed by integrating the Spearman correlation, Pearson correlation, and Cosine similarity matrices, while the gene similarity matrix is derived by computing the Jaccard similarity between genes.

Cell similarity. From each scRNA-seq submatrix, three complementary measures are used to quantify cell-cell similarity: the Pearson correlation coefficient, the Spearman correlation coefficient, and the Cosine similarity. The Pearson similarity matrix is computed as:

(2)

where and denote the expression vector of cells and , respectively. When , the similarity between a cell and itself is set to 1. The Spearman and cosine similarity matrices and are constructed analogously. Next, the three similarity matrices are integrated to calculate the final cell-cell similarity. First, each similarity matrix , is normalized as follows:

(3)

After row-wise min-max normalization, the resulting matrix in Equation (3) is generally asymmetric. We therefore symmetrize it as .

The final cell-cell similarity matrix is obtained by the element-wise geometric mean of the three symmetrized matrices:

(4)

where denotes the number of similarity matrices used.

Gene similarity. We quantify gene–gene similarity with the Jaccard index. The Jaccard similarity coefficient is defined as the ratio of the number of elements at the intersection of two sets and to the number of elements in their union.

In single-cell sequencing data, an expression value greater than zero indicates that the gene is actively expressed in the cell. When two genes are co-expressed at higher levels in a common cell, it suggests that these genes exhibit greater similarity. Initially, we preprocess the gene expression matrix by converting non-zero values to 1, and then calculate the gene similarity matrix by computing the Jaccard similarity coefficient pairwise between genes, as follows:

(5)

where denotes the set of cells in which gene is expressed, and denotes the set of cells in which gene is expressed. If , the similarity of a gene to itself is defined as 1.

Non-linear feature learning module

To extract latent non-linear features from the data, two VGAEs are applied to the cell and gene similarity matrices. Before entering the data into the model, a logarithmic transformation is applied to the submatrix , resulting in the transformed matrix . The VGAE architecture consists of two main components: the encoder and the decoder. In the encoding phase, the first VGAE takes the transpose of the gene similarity matrix and the feature matrix as input, generating a low-dimensional vector representation as output. Similarly, the second VGAE takes the cell similarity matrix and the feature matrix as inputs, producing a low-dimensional vector representation . During the decoding phase, the network parameters are learned using a custom loss function and the reconstruction matrix is obtained based on low-dimensional vector representations and .

The encoder consists of two layers of GCNs [34], defined as follows:

(6)

(7)

where is the input feature matrix, is the input similarity matrix, and .

The encoder is a two-layer : the first layer produces a shared hidden representation, and the second layer splits into two parallel heads— and —that output and , respectively.

The reparameterization trick generates low-dimensional representation vectors while preserving the gradient information for model optimization. It is computed as follows:

(8)

where follows a standard normal distribution .

The decoder is defined by the inner product between the potential low-dimensional vector representations and . The output reconstruction matrix is given by:

(9)

The loss function combines two components: mean squared error (MSE) and Cosine embedding loss. MSE measures the average squared error between the reconstructed matrix and the original feature matrix. Cosine embedding loss assesses the cosine similarity between two vectors against a target similarity label. The formulas are as follows:

(10)

(11)

The joint optimization of MSE and cosine embedding loss is defined as follows:

(12)

where is the weight coefficient that balances each loss function. We perform joint optimization using Adam Optimizer [35]. This optimization yields non-linear representation for cells and for genes.

Linear feature learning module

NMF is applied to the submatrices to extract linear features. For a non-negative matrix , NMF decomposes it into two non-negative matrices and that satisfy the condition [36]. This factorization approximates the original matrix as the product of a cell feature matrix and a gene feature matrix . It is assumed that the gene expression matrix can be well approximated by this factorization. The objective function for NMF is as follows:

(13)

where , . To determine the optimal regularization parameter , we search the following values: . The experimental results indicate that the optimal value of is 0.01.

To optimize the objective, we adopt the alternating-direction method of multipliers algorithm [37] and construct auxiliary functions using Majorization–Minimization and Bregman divergence [38]. The detailed optimization process is described in Supplementary Section 1. Finally, we update and using the multiplicative update rule:

(14)

(15)

We take as input and set the maximum number of iterations to 2000. If the error is less than or the maximum number of iterations is reached, we obtain the linear representation for cells and for genes.

Imputation module

Before using the fully connected layer to predict the features extracted by VGAE and NMF modules, the data is partitioned into training and prediction sets (Fig. 2). First, genes from the raw count matrix are divided into subsets . Spectral clustering is applied to each subset with cluster numbers ranging from 4 to 15, and the three clusters with the highest scores of Silhouette coefficient , , and are selected. Using the submatrix with clusters as an example, we split into . It is hypothesized that within each cell type, there is high similarity between cells and strong correlation in gene expression. For each cluster , if the expression rate of a gene (defined as the proportion of cells in which the gene’s expression level exceeds zero) in exceeds the minimum expression rate observed in both and , the corresponding missing value for that gene is labeled as “1” (false zero). Conversely, if the expression rate of a gene in in this cluster is greater than zero but less than one-tenth of the minimum of the expression rate of and , we mark the missing value of that gene as “0” (representing true biological zeros). The specific conditions for this classification are as follows:

Figure 2.

The process of dividing the training set and prediction set. For each gene group, spectral clustering is applied, and the top three clusters with the largest Silhouette coefficient scores are selected. True and false zeros are then marked for each cluster. The marking results from all clusters are integrated to create a unified set of markers. Finally, the marked matrix is constructed by organizing all the marked gene groups.

(16)

where denotes the expression rate of gene in the cluster of cells and denotes the non-zero rate in the matrix.

To mitigate the effects of spectral clustering variability and reduce over-imputation bias, we apply the aforementioned rule to clusters , , and . The intersection of samples labeled as false zeros across all three clustering results is designated as the prediction set. The training set consists of the intersection of samples labeled as true zeros across the three clustering results and the non-zero samples. After imputation predictions for each submatrix are generated by the fully connected layer, they are merged to form the final imputed matrix.

Experiments

Single-cell RNA sequencing data

We evaluated the performance of our proposed model against several baseline methods on eight real-world scRNA-seq datasets. These datasets encompass a diverse range of species (including mouse and human) and various organs (such as the brain and pancreas). Furthermore, the datasets include cell types such as embryonic stem cells (ES cells) and preimplantation embryos. Detailed descriptions of these datasets are provided in Table 1. The Br_human and Br_mouse datasets are subsets derived from Baron [39].

Table 1.

Summary of eight scRNA-seq datasets used in the experiments

Dataset	Number of cells	Number of genes	Type	Accession	Source (organism/tissue)	Reference
Klein	2717	24 175	4	GSE65525	Mouse ES cells	[40]
Romanov	2881	24 341	7	GSE74672	Mouse brain	[41]
Zeisel	3005	19 972	9	GSE60361	Mouse brain	[42]
Br_human	8569	16 381	14	GSE84133	Human pancreas	[39]
Br_mouse	1886	14 878	13	GSE84133	Mouse pancreas	[39]
Cell type	350	19 097	2	GSE75748	Human ES cells	[43]
Time course	758	19 189	6	GSE75748	Human ES cells	[43]
Petropoulos	1529	16 383	5	E-MTAB-3929	Human preimplantation embryos	[44]

Baseline

We compared the performance of scVGAMF against ten other state-of-the-art imputation methods for scRNA-seq data. These methods include three statistical model-based approaches, two data smoothing techniques, three low-rank matrix methods, and two deep learning-based approaches. A brief overview of these methods is provided in Supplementary Section 2.

Improving the recovery of gene expression

As the true expression levels underlying dropout events in scRNA-seq data are unknown, we simulated the dropout effect by randomly masking non-zero entries. For this experiment, we selected three scRNA-seq datasets: Klein, Romanov, and Zeisel.

First, we preprocessed the raw datasets using the Seurat package in R to filter out genes detected in fewer than 3% of cells. Next, we performed log-normalization and sorted the variable genes to obtain the top 2000 highly variable genes (HVGs). To simulate dropout events, we randomly set non-zero entries to zeros at dropout rates of 10% and 20%, generating a modified expression matrix for each rate.

To evaluate the performance of various imputation methods, we computed the Pearson correlation, cosine similarity, and root mean square error (RMSE) between the imputed values and their corresponding true expression levels. However, as some methods are designed specifically for downstream analytical tasks, it is not always possible to directly obtain the imputed matrix based on the original expression matrix. Consequently, cosine similarity and RMSE comparisons were excluded for these particular methods. For each dataset, we simulated ten independent dropout events. At each dropout rate, we generated boxplots to visualize the Pearson correlation between the imputed and true values for each imputation method (Fig. 3). Cosine similarity and RMSE values were reported as means across the ten experimental replicates (Table 2).

Figure 3.

Evaluation of imputation methods by simulating the dropout effects. (a) Boxplots of the Pearson correlation between the imputed values derived from different methods on the Klein, Romanov and Zeisel datasets. Columns represent masking proportions (10% and 20%). Boxplots show correlation values obtained from 10 simulated replicates. (b) Cosine similarity between the imputed values and real values.

Table 2.

RMSE between the imputed values and real values

	Klein 10%	Klein 20%	Romanov 10%	Romanov 20%	Zeisel 10%	Zeisel 20%
scVGAMF	0.51	0.51	0.65	0.65	0.71	0.71
scRMD	1.02	1.05	1.22	1.25	1.39	1.43
DrImpute	0.82	0.87	0.99	1.03	1.09	1.12
scImpute	0.73	0.75	1.04	1.06	1.08	1.10
TsImpute	0.59	0.63	0.80	0.83	0.86	0.89
CL-Impute	0.71	0.84	0.92	0.95	0.98	1.02

As shown in Fig. 3a, scVGAMF achieved the second-best performance on the Klein dataset at both dropout rates, though its performance was slightly inferior to that of TsImpute. On the Romanov dataset (also evaluated with two dropout rates), scVGAMF consistently ranked among the top three methods, while CL-Impute demonstrated the best performance. On the Zeisel dataset, scVGAMF maintained a second-place ranking, preceded only by CL-Impute. Fig. 3b further showed that scVGAMF ranked second on the Klein dataset, with performance slightly below DrImpute, while outperforming all other methods on both the Romanov and Zeisel datasets. As presented in Table 2, scVGAMF yielded the lowest RMSE values on all three datasets. These results demonstrated that the scVGAMF model achieved high accuracy in predicting gene expression values.

Improving the cell clustering performance

To evaluate the clustering performance, we applied scVGAMF and ten other imputation methods to four real scRNA-seq datasets with known cell type annotations. The datasets used were Br_human, Br_mouse, Romanov, and Zeisel.

Following imputation of the preprocessed gene expression matrices using various scRNA-seq imputation methods, we performed cell clustering on both the imputed datasets and the original raw data using the Seurat [45] package and the SC3 [46] algorithm. To evaluate clustering performance, we computed the Adjusted Rand Index (ARI [47]) and Normalized Mutual Information (NMI [48]) to compare the clustering results with the true cell type labels. Each method was independently run ten times, and the mean ARI and NMI values are reported in Fig. 4.

Figure 4.

Clustering performance evaluation across four real-world datasets. (a) ARI and (b) NMI scores for different clustering methods. Higher values indicate better agreement with ground-truth labels. Error bars represent standard deviation across ten independent runs.

scVGAMF achieved the highest ARI and NMI scores on the Baron_mouse, Romanov, and Zeisel datasets, and ranked second on the Baron_human dataset. Additionally, scVGAMF demonstrated greater stability compared to the other methods. Furthermore, among all the methods evaluated, scVGAMF is the only one that showed an improvement in clustering accuracy across all four datasets compared to the raw dataset, indicating robust performance irrespective of dataset-specific characteristics. Fig. 5 shows the cluster assignment after SC3 was refined with scVGAMF imputation on the Romanov data. With the true cell-type labels as reference, original SC3 erroneously split the cells annotated “1” and “2” into two distinct clusters, while it merged the biologically separate groups “3” and “4” into a single cluster. Both mis-assignments were corrected after imputation.

Figure 5.

UMAP embeddings colored by the ground-truth cell identity (left), SC3 clustering on raw counts (middle), and SC3 clustering on data imputed with scVGAMF (right).

Improving the cell trajectory inference performance

The investigation of cellular dynamics is crucial for both developmental biology and disease research. These dynamic processes can be reconstructed through computational inference of cellular trajectories. However, dropout events result in missing gene expression values, which can significantly compromise the performance of trajectory inference methods. Therefore, accurate imputation of missing values is essential for reliable pseudotime analysis. We employed Monocle 2 [49] to infer pseudotime from raw and imputed data, using default parameters for all analyses. The performance of different imputation methods was assessed by comparing the Pseudo-temporal Ordering Score (POS) and Kendall’s rank correlation coefficient.

We selected two temporal scRNA-seq datasets—Time course and Petropoulos—to validate the performance of the compared methods. The Time course dataset consists of single cells from six stages of human ES cell differentiation (0, 12, 24, 36, 72, and 96 h), while the Petropoulos dataset includes single cells from five stages of human preimplantation embryonic development (Day 3 to Day 7). Each method was independently run ten times, and we calculated the mean POS and Kendall’s rank correlation values. As shown in Fig. 6, scVGAMF achieved the highest POS score (0.888) on the Time course dataset, while both scVGAMF and SAVER ranked first for Kendall’s rank correlation scores (0.719). Fig. 7 shows the POS and Kendall’s rank correlation scores for the Petropoulos dataset. Although scVGAMF achieved median rankings for both POS (0.923) and Kendall’s rank correlation scores (0.71), it significantly outperformed the raw data (POS 0.41, correlation 0.245), demonstrating its capacity to improve trajectory inference accuracy.

Figure 6.

Evaluation of trajectory inference on the Time course dataset. (a) POS scores. (b) Kendall’s rank correlation scores.

Figure 7.

Evaluation of trajectory inference on the Petropoulos dataset. (a) POS scores. (b) Kendall’s rank correlation scores.

Improving the differential expression analysis

Identifying differentially expressed (DE) genes is essential for analyzing scRNA-seq data, as it helps uncover variations in cellular states and provides insight into the underlying molecular mechanisms driving cellular diversity. However, the prevalence of dropout events in scRNA-seq data can significantly compromise the accuracy of differential expression analysis. Because bulk RNA-seq data are less affected by dropout events, we used differential expression results from bulk data as a gold standard to evaluate imputation method performance.

The benchmark dataset from Chu [43] contains both scRNA-seq and bulk RNA-seq data from multiple cell types, including neuronal progenitor cells, definitive endoderm cells (DEC), endothelial cells, trophoblast-like cells, undifferentiated human ES cells (H1 and H9), and human foreskin fibroblasts. Our analysis focused on identifying DE genes between H1 and DEC cells. For the scVGAMF clustering step, we set the number of clusters to range from 2 to 15.

For analysis preparation, we begin by selecting 2000 HVGs from the scRNA-seq data for differential expression analysis. The edgeR package [50] is then employed to identify DE genes, using a maximum false discovery rate of 0.01 and a minimum log fold-change threshold of 1. This approach identified 1160 DE genes in the bulk RNA-seq data.

Fig. 8a shows the Spearman correlation coefficients between adjusted P-values from bulk and imputed (raw) data. The results indicate that scVGAMF identified DE genes with the second-highest consistency compared to the 1160 DE genes from bulk data. In contrast, methods including DrImpute, CL-Impute, SAVER, and ALRA failed to improve Spearman correlation compared to the raw data. We selected the top 300 DE genes based on adjusted -values from bulk data as the gold standard reference. These reference genes were compared with genes identified from raw and imputed data to evaluate the performance of various imputation methods. Fig. 8b illustrates the overlap between reference DE genes from bulk data and those detected in raw versus imputed data. scVGAMF, scImpute, and scRMD were among the top three methods, identifying the largest number of gold-standard DE genes. Additionally, we visualized both raw and imputed data using volcano plots (Fig. 8c). According to Chu [43], DE cells are enriched in genes such as MYCT1, LHX1, NODAL, and PAX6. scVGAMF exhibited higher -log(P) values for these genes compared to the raw data, while maintaining the differential expression of other key genes such as ERBB4, LEFTY1, and DNMT3B. To demonstrate scVGAMF’s potential for enabling scientific discovery, we conducted a systematic biological interpretation of human ES cells differentiation into DECs. The detailed is described in Supplementary Section 3.

Figure 8.

Differential-expression-based evaluation of imputation accuracy. (a) Spearman correlation of adjusted P-values between bulk and single-cell data (raw or imputed). (b) Overlap between single-cell DE genes and the gold-standard set. (c) Volcano plots comparing DE genes in raw data and scVGAMF-imputed data.

Alternative methodologies

Although broad in scope, the downstream evaluations each relied on a single analytical tool. Since each tool has inherent assumptions and limitations, conclusions about scVGAMF’s superiority could be influenced by tool selection. To strengthen the robustness of our comparative results, we implemented alternative analytical pipelines for each task. The results of these alternative methods are provided in Supplementary Section 4.

Ablation study

To evaluate the contribution of non-linear and linear representations, we assessed the performance of the VGAE and NMF components within the scVGAMF framework. Two ablation experiments are conducted for scVGAMF: VGAE (scVGAMF without NMF) and MF (scVGAMF without VGAE). In both ablation experiments, the hyperparameters of the VGAE and MF components are fixed to the same values used in the complete scVGAMF framework. We performed ablation experiments on the Klein, Romanov, and Zeisel datasets, evaluating performance using five metrics (Pearson correlation, Cosine similarity, L1 mean, L1 median, and RMSE). On the Klein dataset, as the simulated dropout rate increased, scVGAMF gradually approached and even outperformed VGAE and MF across all five metrics. In contrast, scVGAMF outperformed MF in all five metrics within the Romanov and Zeisel datasets. The simulated dropout rates and sparsity rate for each dataset are provided in Table 3. These results demonstrate that as dropout rates increase, methods integrating both non-linear and linear features achieve superior performance by mitigating issues associated with high sparsity in single-cell sequencing data.

Table 3.

Results of ablation experiments on the Klein (sparsity rate 60%, 64%, and 68%), Romanov (sparsity rate 82%, 84%, and 86%), and Zeisel (sparsity rate 80%, 82%, and 84%) datasets

	Components	Pearson correlation	Cosine similarity	L1 mean	L1 median	RMSE
	scVGAMF	0.721	0.929	0.405	0.331	0.508
Klein 10%	VGAE	0.579	0.906	0.463	0.387	0.581
	MF	0.723	0.929	0.402	0.328	0.504
	scVGAMF	0.718	0.928	0.406	0.328	0.509
Klein 20%	VGAE	0.568	0.903	0.468	0.394	0.588
	MF	0.72	0.928	0.406	0.331	0.509
	scVGAMF	0.715	0.927	0.411	0.332	0.515
Klein 30%	VGAE	0.551	0.899	0.478	0.404	0.599
	MF	0.715	0.927	0.411	0.334	0.515
	scVGAMF	0.664	0.892	0.502	0.405	0.647
Romanov 10%	VGAE	0.458	0.854	0.589	0.501	0.746
	MF	0.659	0.892	0.502	0.41	0.648
	scVGAMF	0.656	0.89	0.508	0.414	0.654
Romanov 20%	VGAE	0.454	0.852	0.593	0.508	0.75
	MF	0.651	0.89	0.508	0.414	0.654
	scVGAMF	0.654	0.891	0.507	0.414	0.65
Romanov 30%	VGAE	0.444	0.853	0.592	0.508	0.75
	MF	0.646	0.891	0.509	0.424	0.651
	scVGAMF	0.68	0.897	0.543	0.447	0.708
Zeisel 10%	VGAE	0.48	0.857	0.645	0.572	0.822
	MF	0.67	0.895	0.548	0.458	0.714
	scVGAMF	0.672	0.897	0.544	0.449	0.707
Zeisel 20%	VGAE	0.471	0.857	0.646	0.574	0.822
	MF	0.666	0.895	0.548	0.457	0.713
	scVGAMF	0.665	0.895	0.55	0.456	0.714
Zeisel 30%	VGAE	0.464	0.855	0.651	0.579	0.831
	MF	0.657	0.892	0.557	0.47	0.722

Discussion

Dropout events in scRNA-seq data, characterized by high dimensionality, sparsity, and noise, complicate downstream analyses. Technical artifacts often lead to excessive zeros that do not reflect true biological absence, blurring the distinction between technical dropouts and biological zeros. This ambiguity adversely affects critical downstream applications such as clustering, differential expression analysis, and pseudotemporal trajectory inference. Existing imputation methods face two challenges: distinguishing technical zeros from biological zeros, and modeling the mixed linear and non-linear relationships underlying the data. Matrix factorization methods capture linear relationships but miss non-linear patterns, while deep learning methods can model non-linearity but lack interpretability.

To address these issues, we introduce scVGAMF, a hybrid imputation framework that (i) employs consensus spectral clustering to distinguish biological zeros from technical dropouts, (ii) integrates both non-linear and linear features to comprehensively characterize the underlying relationship between cells and genes, and (iii) imputes the dropouts via a fully connected reconstruction layer. Extensive experiments on eight benchmark datasets demonstrate that scVGAMF consistently enhances downstream analyses, including gene–expression recovery, cell clustering, differential gene identification and pseudotemporal ordering. Ablation studies further validate the effectiveness of combining non-linear and linear representations, confirming the synergistic benefits of their fusion.

Beyond accuracy, scVGAMF enhances biological interpretability, improving confidence in cell subtype identification, dynamic expression profiling, and trajectory inference. It provides a robust foundation for future single-cell genomics research.

However, scVGAMF has two limitations: reliance on silhouette coefficients for cluster number selection may not generalize across datasets, and the similarity matrices are sensitive to technical noise. Future work will focus on more elegant strategy, improved noise-resistant similarity estimators, and advanced imputation techniques.

Key Points

• Single-cell variational graph autoencoder and matrix factorization (scVGAMF) integrates linear features (extracted using non-negative matrix factorization) and non-linear features (captured by dual VGAEs), which are then combined by a fully connected neural network to predict missing values.

• Extensive experimental evaluations on eight real single-cell RNA sequencing datasets demonstrate that scVGAMF outperforms ten other imputation methods in gene expression recovery, cell clustering accuracy, differential gene identification, and pseudo-trajectory analysis.

• Ablation studies confirm that the integration of both linear and non-linear features significantly enhances overall imputation performance.

• scVGAMF mitigates the impact of dropout events on downstream analyses while preserving the underlying data structure and correlations.

Author contributions

W.Z. and Y.L. conceived the project and supervised the research. Z.Z. and W.Z. developed and implemented the algorithm. Z.Z, W.Z., J.S., X.Z, and Y.L. analyzed the results. W.Z., Z.Z., and Y.L. wrote the manuscript. All authors read and approved the final paper.

Conflict of interest: The authors declare that they have no competing interests.

Funding

This work was supported by the National Natural Science Foundation (No.12571543, No.12161039 and No.12426649), the Foundation of Wuhan Institute of Technology (No.K2024045), and the Graduate Innovative Fund of Wuhan Institute of Technology (No.CX2024087).

Data availability

The Klein dataset is available at GEO under accession code GSE65525. The Romanov dataset is available at GEO under accession code GSE74672. The Zeisel dataset is available at GEO under accession code GSE60361. The Baron dataset is available at GEO under accession code GSE84133. The Cell type and Time course dataset can be available at GEO under accession code GSE75748. The Petropoulos dataset is available from the ArrayExpress database under accession number E-MTAB-3929. The source code of scVGAMF is available at https://github.com/wzhangwhu/scVGAMF.