Abstract

   Single-cell RNA sequencing (scRNA-seq) enables high-resolution analysis
   of cellular heterogeneity, but dropout events, where gene expression is
   undetected in individual cells, present a significant challenge. We
   propose scMASKGAN, which transforms matrix imputation into a pixel
   restoration task to improve the recovery of missing gene expression
   data. Specifically, we integrate masking, convolutional neural networks
   (CNNs), attention mechanisms, and residual networks (ResNets) to
   effectively address dropout events in scRNA-seq data. The masking
   mechanism ensures the preservation of complete cellular information,
   while convolution and attention mechanisms are employed to capture both
   global and local features. Residual networks augment feature
   representation and effectively mitigate the risk of model overfitting.
   Additionally, cell-type labels are incorporated as constraints to guide
   the methods in learning more accurate cellular features. Finally,
   multiple experiments were conducted to evaluate the methods’
   performance using seven different data types and scRNA-seq data from
   ten neuroblastoma samples. The results demonstrate that the data
   imputed by scMASKGAN not only perform excellently across various
   evaluation metrics but also significantly enhance the effectiveness of
   downstream analyses, enabling a more comprehensive exploration of
   underlying biological information.

Supplementary Information

   The online version contains supplementary material available at
   10.1186/s12859-025-06138-9.

   Keywords: Single-cell RNA sequencing, Data imputation, Deep learning

Introduction

   Single-cell RNA sequencing (scRNA-seq) measures gene expression at the
   single-cell level, providing valuable insights into cellular
   heterogeneity and diversity [[32]1, [33]2]. This technique involves
   reverse transcription and amplification of full-length mRNA [[34]3],
   but typically focuses on the 5’ and 3’ ends, introducing inherent
   limitations [[35]4]. Variability in reverse transcription efficiency
   and limited starting RNA contribute to high technical noise, capturing
   only a fraction of the transcriptome and generating many zero
   expression values [[36]5]. These zeros may represent either true lack
   of expression or artificial dropout events caused by technical noise
   [[37]6], obscuring the true gene expression landscape. Consequently,
   advanced methods are needed to accurately identify and mitigate dropout
   events in scRNA-seq analysis.

   Early approaches to address this challenge include methods such as
   MAGIC [[38]7], SAVER [[39]8], DrImpute [[40]9], VIPER [[41]10],
   scImpute [[42]11], ENHANCE [[43]12], and SCRABBLE [[44]13]. MAGIC
   imputes missing values by leveraging similarities between cells, while
   SAVER uses a Bayesian framework for probabilistic data recovery.
   DrImpute groups similar cells via clustering for imputation, and VIPER
   employs a non-negative sparse regression model based on neighborhood
   information. Additionally, ENHANCE and SCRABBLE utilize principal
   component analysis and batch normalization techniques, respectively, to
   reduce dimensionality and mitigate batch effects, thereby enhancing
   data recovery quality.

   Recent years have seen a growing application of deep learning methods
   to address dropout events in scRNA-seq data [[45]14]. Several
   approaches-including DeepImpute [[46]15], AutoImpute [[47]16], DCA
   [[48]17], scVI [[49]18, [50]19], DISC [[51]20], and sciGANs
   [[52]21]-have made significant strides in this domain. For example,
   DeepImpute employs a divide-and-conquer strategy with sub-neural
   networks to estimate gene expression, outperforming earlier methods
   such as DrImpute and SAVER. However, its reliance on extensive
   parameter tuning and using 95
   [MATH: <mo>%</mo> :MATH]
   of the data for training can lead to overfitting. AutoImpute learns the
   inherent data distribution for imputation but has been observed to
   sometimes produce invalid (negative) values. DCA enhances traditional
   autoencoders by incorporating negative binomial and zero-inflated noise
   models [[53]22], thereby improving both denoising and imputation. scVI,
   a hierarchical Bayesian model, addresses batch correction and
   imputation by projecting data into latent spaces, although it may
   struggle when gene counts exceed cell numbers. DISC improves
   reliability by training on datasets with a high proportion of
   unexpressed genes (90
   [MATH: <mo>%</mo> :MATH]
   ), but its efficiency is constrained by the need for high-performance
   hardware. sciGANs leverages Generative Adversarial Networks (GANs)
   [[54]23] to frame dropout imputation as a pixel recovery task, thus
   avoiding overfitting and maintaining robustness in detecting
   low-expression genes, but converting each row of scRNA-seq data into a
   square image matrix can introduce significant noise.

   Earlier imputation algorithms, such as MAGIC and SAVER, are typically
   based on assumptions of cell similarity or gene co-expression networks.
   While these methods can enhance signal clarity, they tend to remove
   lowly expressed genes and overlook the inherent stochasticity of
   natural cellular processes, thereby complicating the imputation of rare
   cell populations. More recent deep learning-based approaches (e.g.,
   scIALM [[55]24], SAE-Impute [[56]25]) often incorporate mathematical
   assumptions and regularization techniques to constrain the imputation
   process. Although these methods may improve imputation accuracy or
   enhance the visual quality of UMAP clustering plots, they frequently
   produce excessively smoothed results that compromise the representation
   of true biological variability. In contrast, GAN-based approaches can
   effectively circumvent these limitations by reframing imputation as a
   generative process. This strategy allows GANs to learn the true data
   distribution, generate realistic synthetic data points, and use these
   generated samples to estimate missing values, thereby mitigating the
   risk of overfitting while better preserving the intrinsic biological
   variability.

   In this study, we introduce scMASKGAN, a GAN-based framework for
   scRNA-seq data imputation that combines masking, convolutional neural
   networks (CNN) [[57]26], attention mechanisms [[58]27], and ResNets
   [[59]28] to effectively address dropout events. Our main contributions
   are summarized below:
     * We design scMASKGAN with a novel masking mechanism that preserves
       the intrinsic structure of gene expression data without imposing
       gene-specific constraints. By integrating CNNs, Self-attention, and
       ResNets, the model dynamically captures intricate gene-gene and
       gene-cell interactions, learning hierarchical representations that
       maintain both local and global biological features.
     * Instead of directly modifying the original data, scMASKGAN
       generates realistic synthetic single-cell data for imputation,
       thereby avoiding overfitting to dominant cell types while retaining
       features of rare cells. Additionally, an Isolation Forest algorithm
       is employed to detect and remove anomalous values during synthesis,
       ensuring high biological fidelity in the final imputed dataset.
     * We validate scMASKGAN on seven diverse datasets and 10
       neuroblastoma samples, demonstrating its superior performance over
       existing imputation methods across multiple quantitative metrics,
       as well as its ability to restore biologically meaningful gene
       expression patterns, including improved gene-gene correlations,
       trajectory inference, batch correction, and differential expression
       analysis.

Materials and methods

Data preparation

   We utilize seven real datasets from Xu et al. [[60]21], encompassing a
   diverse array of scRNA-seq data types. These datasets include Human
   brain scRNA-seq data,[61]1 ERCC spike-in RNAs scRNA-seq data[62]2,
   Mouse ESC scRNA-seq data,[63]3 time-course scRNA-seq data,[64]4 and
   three scRNA-seq datasets derived from the sc_Drop-seq,[65]5
   sc_CEL-seq2,[66]6 and sc_10X platforms.[67]7 The details is as follows:
     * Human brain scRNA-seq data: This dataset provides high-quality
       single-cell RNA sequencing data from human brain tissues. It is
       used to evaluate the performance of imputation methods in complex
       biological systems with diverse cell types and intricate gene
       expression patterns.
     * ERCC spike-in RNAs scRNA-seq data: This dataset is unique in that
       it consists of spike-in RNA molecules, where the number of cells
       exceeds the number of genes. It serves as a benchmark for
       evaluating the imputation performance on extremely small gene
       expression matrix datasets.
     * Mouse ESC scRNA-seq data: This dataset includes embryonic stem cell
       (ESC) differentiation and large-scale scRNA-seq data. It allows us
       to test the imputation method’s effectiveness in developmental
       biology studies, where gene expression profiles exhibit dynamic
       changes.
     * Time-course scRNA-seq data: Temporal sequencing data is crucial for
       understanding gene expression dynamics over time. This dataset
       helps evaluate how well the imputation method preserves temporal
       patterns and recovers missing values in time-dependent scRNA-seq
       studies.
     * sc_Drop-seq dataset: Drop-seq is a widely used droplet-based
       scRNA-seq technology. By including this dataset, we assess the
       robustness of scMASKGAN across different sequencing platforms,
       ensuring its compatibility with droplet-based data.
     * sc_CEL-seq2 dataset: CEL-seq2 is a plate-based sequencing platform
       known for its improved sensitivity and accuracy in capturing gene
       expression. This dataset helps evaluate the imputation model’s
       adaptability to high-precision sequencing techniques.
     * sc_10X dataset: The 10X Genomics platform is one of the most widely
       used commercial scRNA-seq technologies. Including this dataset
       ensures that scMASKGAN can generalize to large-scale single-cell
       datasets generated from commercial platforms, demonstrating its
       scalability and broad applicability.

   By incorporating datasets from different species, sequencing platforms,
   data sizes, dropout rate and experimental conditions, we ensure that
   our evaluation of scMASKGAN is comprehensive and reflects its potential
   utility in diverse real-world applications. Detailed information on
   these datasets is provided in Table [68]1.

Table 1.

   Seven scRNA-seq samples
   Datasets Type Cells Genes Accession number Dropout
   Brain Human 420 22085 [69]GSE67835 81
   [MATH: <mo>%</mo> :MATH]
   ERCC Spike-in 288 92 E-MTAB-2512 33
   [MATH: <mo>%</mo> :MATH]
   ESC Mouse 6886 24176 [70]GSE65525 70
   [MATH: <mo>%</mo> :MATH]
   sc_Drop-seq ... 226 15128 [71]GSE118706 62
   [MATH: <mo>%</mo> :MATH]
   sc_CEL-seq2 ... 275 28205 [72]GSE117617 74
   [MATH: <mo>%</mo> :MATH]
   sc_10X ... 740 16469 [73]GSE111108 45
   [MATH: <mo>%</mo> :MATH]
   Time-course ... 759 19190 [74]GSE75748 55
   [MATH: <mo>%</mo> :MATH]
   [75]Open in a new tab

   Additionally, we employ a dataset from Yuan et al. [[76]29] that
   comprises 10 neuroblastoma samples, including 5 high-risk
   neuroblastomas (NB) and 5 low-risk neuroblastomas-among which there are
   4 ganglioneuroblastomas (GNB) and 1 ganglioneuroma (GN). These data can
   be downloaded from GEO,[77]8 as detailed in Table [78]2. The samples
   were obtained from clinical surgical resections and belong to the same
   sequencing batch, which facilitates a comparative assessment of model
   performance across batches and ensures that the imputation process
   mitigates batch effects during data integration. Moreover, as dropout
   rates exceed 90% in most of these datasets, it facilitates rigorous
   testing of the model’s robustness under extreme conditions. In
   subsequent figures, we will primarily use the dropout rates of
   different datasets as legends or titles.

Table 2.

   Ten neuroblastoma samples
   Datasets Type Cells Genes Dropout
   [79]GSM5768743 NB 960 33514 95
   [MATH: <mo>%</mo> :MATH]
   [80]GSM5768744 NB 768 33514 91
   [MATH: <mo>%</mo> :MATH]
   [81]GSM5768745 NB 445 33514 97
   [MATH: <mo>%</mo> :MATH]
   [82]GSM5768746 NB 357 33514 85
   [MATH: <mo>%</mo> :MATH]
   [83]GSM5768747 NB 639 33514 96
   [MATH: <mo>%</mo> :MATH]
   [84]GSM5768748 GNB 740 33514 97
   [MATH: <mo>%</mo> :MATH]
   [85]GSM5768749 GNB 1052 33514 97
   [MATH: <mo>%</mo> :MATH]
   [86]GSM5768750 GNB 1053 33514 97
   [MATH: <mo>%</mo> :MATH]
   [87]GSM5768751 GNB 360 33514 84
   [MATH: <mo>%</mo> :MATH]
   [88]GSM5768752 GN 551 33514 97
   [MATH: <mo>%</mo> :MATH]
   [89]Open in a new tab

Methods

   Inspired by the remarkable performance of GANs in image restoration and
   the introduction of StackGAN [[90]30], a model designed for
   text-to-image generation, we propose scMASKGAN, a generative
   adversarial network tailored for scRNA-seq imputation. Both image
   inpainting and missing value imputation share the fundamental goal of
   restoring incomplete data while preserving its structural and
   contextual integrity. In image inpainting, GANs learn spatial
   structures and content features to generate realistic pixels that
   seamlessly blend with surrounding regions. Similarly, in missing value
   imputation, GANs leverage gene expression patterns and intercellular
   relationships to generate biologically meaningful values that
   accurately reflect the underlying data distribution. By converting each
   cell’s normalized gene expression profile into a grayscale image,
   scMASKGAN extracts essential features and utilizes GANs’ robust image
   synthesis capabilities to generate realistic grayscale representations
   that impute missing gene data. The detailed framework is shown in
   Fig. [91]1, and in this section, we provide a comprehensive description
   of the scMASKGAN workflow.

Fig. 1.

   [92]Fig. 1
   [93]Open in a new tab

   The overall framework of scMASKGAN is illustrated across four panels.
   Panel A shows the process of preparing data, beginning with the
   sequencing-derived gene expression matrix and converting it into a
   format suitable for model input. Panel B details the architecture of
   discriminator, designed to distinguish between original and imputed
   data, thereby driving the generator to produce more biologically
   accurate imputed values, the upsampling module is designed to increase
   data dimensionality and capture detailed distribution features. The
   downsampling module reduces resolution and utilizes convolution to
   extract higher-order features. Residual blocks are employed to enhance
   these high-level features, while the attention module is used to
   capture global characteristics. Panel C presents the architecture of
   generator, which uses multiple layers and mechanisms to infer and fill
   missing values based on the observed data patterns. Panel D outlines
   the complete imputation workflow

Data preprocessing

   scMASKGAN reframes the imputation of single-cell RNA-seq data as an
   image inpainting (pixel restoration) problem by converting the gene
   expression matrix into an image-based representation. As shown in
   Fig. [94]1A, each cell’s expression profile (originally a vector of m
   gene features) is mapped onto a 2D grid of pixels of size
   [MATH: <mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow> :MATH]
   (chosen such that
   [MATH: <mrow><mi>N</mi><mo>×</mo><mi>N</mi><mo>≥</mo><mi>m</mi></mrow>
   :MATH]
   , ). Specifically, the gene expression vector of each cell is
   zero-padded to a length of
   [MATH: <mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow> :MATH]
   , reshaped into a two-dimensional array by sequentially arranging the
   elements row-wise, thus ensuring consistency with the original matrix.
   In this image, each pixel corresponds to a specific gene, and its
   intensity represents the expression level of that gene in the given
   cell. For a gene j mapped to pixel coordinate (u, v), we have:
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><msub><mi>I</mi><mi>i</mi></msub><mrow><mo
   stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>x</mi><mrow><mi>i</mi
   ><mo>,</mo><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow>
   :MATH]
   1

   where
   [MATH:
   <msub><mi>x</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub>
   :MATH]
   denotes the expression value of gene j in cell i. Dropout events (genes
   with undetected expression in a cell) manifest as pixels with zero
   intensity in the image, and scMASKGAN applies a masking mechanism to
   flag these missing values. A binary mask of the same
   [MATH: <mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow> :MATH]
   size is generated for each cell, using 0 for dropouts and 1 for
   observed gene expressions. This masking mechanism serves two critical
   functions. First, since gene expression values are initially
   normalized, missing entries would otherwise be assigned nonzero values,
   potentially introducing noise when extracting features using CNNs and
   attention mechanisms. The binary mask effectively mitigates this
   interference by distinguishing observed values from imputed ones.
   Second, by masking the padded values, the model ensures that they do
   not participate in the computation process, preserving the integrity of
   the learned representations. This image-based representation allows
   missing data imputation to be treated as a pixel-wise image restoration
   task, enabling convolutional neural networks with attention mechanisms
   to capture both local gene-gene interaction patterns and global
   expression structures from the spatial layout, leading to more
   effective recovery of missing data.

   Moreover, cell-type labels (
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   ) is extracted from the dataset and converted into categorical
   numerical values to represent different cell types. In scMASKGAN
   training, the label is transformed into a one-hot encoded vector and
   incorporated into both the generator and discriminator, ensuring
   class-specific data generation and enhancing classification
   performance. In the imputation process, label is utilized to associate
   each cell with the corresponding generated candidate set, thereby
   ensuring that missing values are imputed using biologically relevant
   data from the same category.

Generator

   In scMASKGAN, the generator
   [MATH: <msub><mi>G</mi><msub><mi>θ</mi><mi>G</mi></msub></msub> :MATH]
   transforms a random noise vector
   [MATH: <mi mathvariant="bold">z</mi> :MATH]
   into a synthetic gene expression tensor, conditioned on a one-hot
   encoded label matrix
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   through a series of linear and non-linear operations, as illustrated in
   Fig. [95]1B. Initially, the noise vector
   [MATH: <mrow><mi mathvariant="bold">z</mi><mo>∈</mo><msup><mrow><mi
   mathvariant="double-struck">R</mi></mrow><msub><mi>d</mi><mi>z</mi></ms
   ub></msup></mrow> :MATH]
   is sampled from a prior distribution
   [MATH: <msub><mi>p</mi><mi>z</mi></msub> :MATH]
   , (
   [MATH: <mrow><mi mathvariant="bold">z</mi><mo>∼</mo><mi
   mathvariant="script">N</mi><mo stretchy="false">(</mo><mn
   mathvariant="bold">0</mn><mo>,</mo><mn mathvariant="bold">1</mn><mo
   stretchy="false">)</mo></mrow> :MATH]
   ). The generator’s architecture is defined by the following sequence of
   transformations:
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><msub><mi>G</mi><msub><mi>θ</mi><mi>G</mi></m
   sub></msub><mo>:</mo><mspace width="0.277778em"></mspace><mi
   mathvariant="bold">z</mi><mo>,</mo><mi
   mathvariant="bold">y</mi><mover><mo
   stretchy="false">→</mo><msub><mi>F</mi><mn>1</mn></msub></mover><msup><
   mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msup><mover><mo
   stretchy="false">→</mo><msub><mi>F</mi><mn>2</mn></msub></mover><msup><
   mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>2</mn><mo
   stretchy="false">)</mo></mrow></msup><mover><mo
   stretchy="false">→</mo><mrow><msub><mi>F</mi><mn>3</mn></msub><mo>,</mo
   ><mo>⋯</mo><mo>,</mo><msub><mi>F</mi><mi>L</mi></msub></mrow></mover><m
   sup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mi>L</mi><mo
   stretchy="false">)</mo></mrow></msup><mo>=</mo><mi>G</mi><mrow><mo
   stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo>,</mo><mi
   mathvariant="bold">y</mi><mo
   stretchy="false">)</mo></mrow></mrow></mtd></mtr></mtable></mrow>
   :MATH]
   2

   where each
   [MATH: <msub><mi>F</mi><mi>l</mi></msub> :MATH]
   denotes a parameterized transformation (either linear or
   convolutional), followed by a ReLU or Sigmoid activation function to
   ensure the non-negativity of scRNA-seq data and the binary
   representation of grayscale images. The final output
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mi>L</mi><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   constitutes the generated tensor, which represents imputed gene
   expression data. The objective of the generator is to produce synthetic
   data that conform to the biological context specified by the label
   matrix
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   . The generator is optimized by minimizing the following loss function:
   [MATH: <mrow><mtable><mtr><mtd columnalign="right"><mrow><munder><mo
   movablelimits="true">min</mo><mi>G</mi></munder><mspace
   width="0.277778em"></mspace><msub><mi
   mathvariant="double-struck">E</mi><mrow><mi
   mathvariant="bold">z</mi><mo>∼</mo><msub><mi>p</mi><mi>z</mi></msub><mr
   ow><mo stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo
   stretchy="false">)</mo></mrow></mrow></msub><mfenced close="]"
   open="["><mo>log</mo><mfenced close=")"
   open="("><mn>1</mn><mo>-</mo><mi>D</mi><mfenced close=")"
   open="("><mi>G</mi><mo stretchy="false">(</mo><mi
   mathvariant="bold">z</mi><mo>∣</mo><mi mathvariant="bold">y</mi><mo
   stretchy="false">)</mo><mo>∣</mo><mi
   mathvariant="bold">y</mi></mfenced></mfenced></mfenced></mrow></mtd></m
   tr></mtable></mrow> :MATH]
   3

   where
   [MATH: <mi>D</mi> :MATH]
   represents the discriminator and
   [MATH: <mrow><mi>G</mi><mo stretchy="false">(</mo><mi
   mathvariant="bold">z</mi><mo>∣</mo><mi mathvariant="bold">y</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   is the generator output conditioned on
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   . This loss encourages the generator to produce outputs that are
   indistinguishable from real data under the given cell-type conditions.
   Initially, the generator applies a linear transformation, batch
   normalization, and ReLU activation to the input noise vector
   [MATH: <mi mathvariant="bold">z</mi> :MATH]
   , yielding an intermediate feature map
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>1</mn><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   . This feature map is subsequently reshaped into a tensor of dimensions
   [MATH: <mfenced close=")"
   open="("><mi>B</mi><mo>,</mo><mtext>cn1</mtext><mo>,</mo><mfrac><mtext>
   img\_size</mtext><mn>4</mn></mfrac><mo>,</mo><mfrac><mtext>img\_size</m
   text><mn>4</mn></mfrac></mfenced> :MATH]
   , where
   [MATH: <mi>B</mi> :MATH]
   is the batch size and
   [MATH: <mtext>cn1</mtext> :MATH]
   denotes the number of channels in the first convolutional block. Next,
   a self-attention mechanism is employed to capture long-range
   dependencies among features. In this mechanism, attention scores are
   computed by learning queries
   [MATH: <mi mathvariant="bold">Q</mi> :MATH]
   , keys
   [MATH: <mi mathvariant="bold">K</mi> :MATH]
   , and values
   [MATH: <mi mathvariant="bold">V</mi> :MATH]
   , and the attention output
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mtext>att</mtext><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   is calculated as follows:
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><mtext>Attention</mtext><mo>=</mo><mtext>Soft
   max</mtext><mfenced close=")" open="("><mfrac><mrow><mi
   mathvariant="bold">Q</mi><msup><mi
   mathvariant="bold">K</mi><mi>⊤</mi></msup></mrow><msqrt><mi>d</mi></msq
   rt></mfrac></mfenced><mo>,</mo><mspace width="1em"></mspace><msup><mi
   mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mtext>att</mtext><mo
   stretchy="false">)</mo></mrow></msup><mo>=</mo><mi
   mathvariant="bold">V</mi><mo>·</mo><mtext>Attention</mtext><mo>.</mo></
   mrow></mtd></mtr></mtable></mrow> :MATH]
   4

   This output is integrated with the preceding feature map via a residual
   connection, modulated by a learnable scaling parameter
   [MATH: <mi>γ</mi> :MATH]
   . Concurrently, the label matrix
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   is upsampled to match the spatial dimensions of the feature map
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>3</mn><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   and processed through a convolutional layer to reduce its channel
   dimensionality. The resulting label-aligned feature map is then
   concatenated with
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>3</mn><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   to form the final feature map
   [MATH: <msup><mi mathvariant="bold">H</mi><mrow><mo
   stretchy="false">(</mo><mn>5</mn><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   , which is further processed by additional convolutional layers. The
   output image
   [MATH: <mrow><mi>G</mi><mo stretchy="false">(</mo><mi
   mathvariant="bold">z</mi><mo>,</mo><mi mathvariant="bold">y</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   is obtained by applying a sigmoid activation to the final convolutional
   layer’s output. The detailed design of the generator is as follows:

Algorithm 1.

   [96]Algorithm 1
   [97]Open in a new tab

   scMASKGAN Generator

Discriminator

   The primary function of the discriminator
   [MATH: <mi>D</mi> :MATH]
   is to distinguish between real and synthetic gene expression data. As
   illustrated in Fig. [98]1C, the discriminator receives as input an
   image
   [MATH: <mi>x</mi> :MATH]
   representing gene expression data and a label matrix
   [MATH: <mi>y</mi> :MATH]
   encoding cell-type information. Its optimization objective is defined
   as
   [MATH: <mrow><mtable><mtr><mtd columnalign="right"><mrow><munder><mo
   movablelimits="true">max</mo><mi>D</mi></munder><mspace
   width="0.277778em"></mspace><msub><mi
   mathvariant="double-struck">E</mi><mrow><mi>x</mi><mo>∼</mo><msub><mi>p
   </mi><mtext>data</mtext></msub><mrow><mo
   stretchy="false">(</mo><mi>x</mi><mo
   stretchy="false">)</mo></mrow></mrow></msub><mfenced close="]"
   open="["><mo>log</mo><mi>D</mi><mo
   stretchy="false">(</mo><mi>x</mi><mo>∣</mo><mi>y</mi><mo
   stretchy="false">)</mo></mfenced><mo>+</mo><msub><mi
   mathvariant="double-struck">E</mi><mrow><mi>z</mi><mo>∼</mo><msub><mi>p
   </mi><mi>z</mi></msub><mrow><mo stretchy="false">(</mo><mi>z</mi><mo
   stretchy="false">)</mo></mrow></mrow></msub><mfenced close="]"
   open="["><mo>log</mo><mrow><mo maxsize="1.623em" minsize="1.623em"
   stretchy="true">(</mo></mrow><mn>1</mn><mo>-</mo><mi>D</mi><mrow><mo
   maxsize="1.2em" minsize="1.2em"
   stretchy="true">(</mo></mrow><mi>G</mi><mo
   stretchy="false">(</mo><mi>z</mi><mo>∣</mo><mi>y</mi><mo
   stretchy="false">)</mo><mo>∣</mo><mi>y</mi><mrow><mo maxsize="1.2em"
   minsize="1.2em" stretchy="true">)</mo></mrow><mrow><mo
   maxsize="1.623em" minsize="1.623em"
   stretchy="true">)</mo></mrow></mfenced></mrow></mtd></mtr></mtable></mr
   ow> :MATH]
   5

   where
   [MATH: <mrow><mi>D</mi><mo
   stretchy="false">(</mo><mi>x</mi><mo>∣</mo><mi>y</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   denotes the probability that the discriminator assigns to a real sample
   [MATH: <mi>x</mi> :MATH]
   under the condition
   [MATH: <mi>y</mi> :MATH]
   , and
   [MATH: <mrow><mi>D</mi><mo stretchy="false">(</mo><mi>G</mi><mo
   stretchy="false">(</mo><mi>z</mi><mo>∣</mo><mi>y</mi><mo
   stretchy="false">)</mo><mo>∣</mo><mi>y</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   corresponds to the probability assigned to a generated sample.

   Specifically, the input image
   [MATH: <mi>x</mi> :MATH]
   is first processed by a preprocessing layer that flattens the image and
   reconstructs it into a feature map suitable for convolutional
   operations. This feature map is then passed through a downsampling
   convolutional block to reduce its spatial dimensions, with residual
   connections incorporated to enhance feature propagation and facilitate
   model learning. Moreover, a self-attention mechanism is integrated to
   improve feature extraction by computing attention scores and generating
   the corresponding output
   [MATH: <msup><mi>H</mi><mrow><mo
   stretchy="false">(</mo><mtext>att</mtext><mo
   stretchy="false">)</mo></mrow></msup> :MATH]
   . Simultaneously, the label matrix
   [MATH: <mi>y</mi> :MATH]
   is upsampled and processed analogously to the procedure employed in the
   generator, after which it is concatenated with the image features to
   form a label-aligned feature map. Finally, a series of upsampling
   layers is applied to restore the spatial dimensions of the feature map,
   ultimately yielding a probability map that reflects the authenticity of
   the input image
   [MATH: <mi>x</mi> :MATH]
   under the condition
   [MATH: <mi>y</mi> :MATH]
   . The detailed design of the discriminator is as follows:

Algorithm 2.

   [99]Algorithm 2
   [100]Open in a new tab

   scMASKGAN Discriminator

   Overall, the generator and discriminator collaborate within an
   adversarial framework. The generator produces realistic gene expression
   data conditioned on the label matrix
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   . The discriminator distinguishes between real and generated data based
   on the same cell-type label conditioning. This architecture enables the
   generation of realistic gene expression profiles. The detailed design
   of the scMASKGAN is as follows:

Algorithm 3.

   [101]Algorithm 3
   [102]Open in a new tab

   scMASKGAN Training Process

Imputed method

   For a given cell
   [MATH: <msub><mi>c</mi><mi>i</mi></msub> :MATH]
   in subgroup
   [MATH: <msub><mi>K</mi><msub><mi>c</mi><mi>i</mi></msub></msub> :MATH]
   , we first generate a candidate set
   [MATH:
   <msub><mi>A</mi><msub><mi>K</mi><msub><mi>c</mi><mi>i</mi></msub></msub
   ></msub> :MATH]
   of
   [MATH: <msub><mi>n</mi><mtext>can</mtext></msub> :MATH]
   expression profiles using a well-trained scMASKGAN. Specifically, the
   one-hot encoded cell-type vectors are combined with randomly sampled
   noise and fed into the pre-trained generator. Through multiple
   transformations and mapping processes, the generator produces a
   candidate set of gene expression profiles that correspond to the given
   cell type. We then apply the Isolation Forest (ISOforest)
   algorithm[[103]31] (Fig. [104]1D) to detect and remove outliers by
   constructing isolation trees and assigning anomaly scores based on path
   lengths-where shorter paths indicate potential outliers. Specifically,
   each tree recursively partitions the data by randomly selecting
   features and split points. The depth of an isolation tree, which
   represents the number of splits required to isolate a data point,
   reflects the difficulty of isolating that point. To quantify the degree
   of anomaly, an anomaly score
   [MATH: <mrow><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   is computed for each candidate gene expression profile using the
   formula
   [MATH: <mrow><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><msup><mn>2</mn><mrow><mo>-</mo
   ><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo
   stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo
   stretchy="false">)</mo></mrow><mrow><mi>c</mi><mo
   stretchy="false">(</mo><mi>n</mi><mo
   stretchy="false">)</mo></mrow></mfrac></mrow></msup></mrow> :MATH]
   . where
   [MATH: <mrow><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo
   stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo
   stretchy="false">)</mo></mrow> :MATH]
   denotes the average path length of data point
   [MATH: <mi>x</mi> :MATH]
   across all isolation trees, and
   [MATH: <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>n</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   represents the average path length for a dataset of size
   [MATH: <mi>n</mi> :MATH]
   . Since anomalous points are generally easier to isolate, they tend to
   have shorter path lengths and, consequently, higher anomaly scores.
   Finally, a threshold
   [MATH: <mi>τ</mi> :MATH]
   is set to identify outliers, where a data point is classified as an
   anomaly if
   [MATH: <mrow><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo
   stretchy="false">)</mo><mo>></mo><mi>τ</mi></mrow> :MATH]
   The threshold
   [MATH: <mi>τ</mi> :MATH]
   can be dynamically determined based on the distribution of anomaly
   scores within the dataset. Moreover, we find scMASKGAN employs the
   Isolation Forest anomaly detection algorithm to identify and remove
   outliers during the imputation of batch data in Sect. [105]4.3. By
   eliminating anomalies introduced during batch sequencing, this approach
   reduces the impact of batch effects.

   Finally, using Euclidean distance (
   [MATH: <mrow><mi>d</mi><mrow><mo stretchy="false">(</mo><mi
   mathvariant="bold">x</mi><mo>,</mo><mi mathvariant="bold">y</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><msqrt><mrow><msubsup><mo>∑</mo
   ><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><
   mrow><mo
   stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><msub
   ><mi>y</mi><mi>i</mi></msub><mo
   stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow>
   :MATH]
   ), where
   [MATH: <mi mathvariant="bold">x</mi> :MATH]
   and
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   represent the gene expression vectors of two cells, with a dimension of
   [MATH: <mi>n</mi> :MATH]
   (i.e., the number of genes). To ensure computational accuracy and
   comparability, all gene expression data undergo normalization
   preprocessing before computing the Euclidean distance, thereby
   eliminating the influence of varying gene expression ranges.
   Furthermore, to enhance the reliability of nearest neighbor selection,
   we define a dynamically adjusted similarity threshold
   [MATH: <mi>θ</mi> :MATH]
   , which is determined based on the statistical distribution of
   Euclidean distances between all cell pairs. Typically, it is set as the
   mean
   [MATH: <mi>μ</mi> :MATH]
   plus a certain multiple of the standard deviation
   [MATH: <mi>σ</mi> :MATH]
   , i.e.,
   [MATH:
   <mrow><mi>θ</mi><mo>=</mo><mi>μ</mi><mo>+</mo><mi>k</mi><mi>σ</mi></mro
   w> :MATH]
   . To optimize the imputation performance, we further employ Optuna
   [[106]32] for hyperparameter tuning, primarily optimizing the
   [MATH: <mi>k</mi> :MATH]
   -nearest neighbors (KNN) parameter
   [MATH: <mi>k</mi> :MATH]
   . The objective is to search for the optimal
   [MATH: <mi>k</mi> :MATH]
   value that minimizes the mean squared error (MSE) between the original
   and imputed data, ensuring that selected neighbors exhibit high
   similarity while avoiding the influence of outliers. Ultimately, only
   cells with a Euclidean distance less than
   [MATH: <mi>θ</mi> :MATH]
   are retained as candidate neighbors, improving the reliability and
   biological interpretability of the imputed data. To estimate the
   expression value of gene
   [MATH: <mi>j</mi> :MATH]
   in cell
   [MATH: <msub><mi>c</mi><mi>i</mi></msub> :MATH]
   , we use the following formula:
   [MATH: <mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mover
   accent="true"><mi>c</mi><mo
   stretchy="false">^</mo></mover><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mr
   ow></msub><mo>=</mo><mfenced open="{"><mrow><mtable><mtr><mtd
   columnalign="left"><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo>,</mo><mi>
   j</mi></mrow></msub><mo>,</mo></mrow></mtd><mtd
   columnalign="left"><mrow><mtext>if</mtext><mspace
   width="0.333333em"></mspace><msub><mi>c</mi><mrow><mi>i</mi><mo>,</mo><
   mi>j</mi></mrow></msub><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr
   ><mtr><mtd
   columnalign="left"><mrow><mrow></mrow><msub><mi>c</mi><mrow><mi>i</mi><
   mo>,</mo><mi>k</mi><mi>N</mi><mi>N</mi><mo>,</mo><mi>j</mi></mrow></msu
   b><mo>,</mo></mrow></mtd><mtd
   columnalign="left"><mrow><mtext>otherwise</mtext><mo>.</mo></mrow></mtd
   ></mtr></mtable></mrow></mfenced></mrow></mtd></mtr></mtable></mrow>
   :MATH]
   6

   Here,
   [MATH: <msub><mover accent="true"><mi>c</mi><mo
   stretchy="false">^</mo></mover><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mr
   ow></msub> :MATH]
   denotes the estimated expression value,
   [MATH:
   <msub><mi>c</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub>
   :MATH]
   is the original expression value from the raw profile, and
   [MATH:
   <msub><mi>c</mi><mrow><mi>i</mi><mo>,</mo><mi>k</mi><mi>N</mi><mi>N</mi
   ><mo>,</mo><mi>j</mi></mrow></msub> :MATH]
   represents the value estimated from the expression profiles of the
   nearest neighbor cells. If the expression value of gene
   [MATH: <mi>j</mi> :MATH]
   in cell
   [MATH: <msub><mi>c</mi><mi>i</mi></msub> :MATH]
   is greater than zero, the original value is retained. Otherwise, the
   corresponding gene expression value from the nearest neighbors is used
   for imputation.

   Finally, we obtained a matrix consisting of the imputed scRNA-seq data
   along with padded zeros. Since we applied a masking mechanism that
   ignored these zeros, they were not processed during model training and
   imputation. Therefore, by removing the padded zeros, we obtained the
   final imputed matrix.

Parameter adjustment strategies

   The parameter-tuning strategy involves carefully adjusting the learning
   rate (lr), typically starting from values such as 0.001 or 0.0001, and
   then employing Adam [[107]33] with weighted iterative refinements based
   on observed convergence performance. The parameter
   [MATH: <mtext>img\_size</mtext> :MATH]
   , defined as dimension
   [MATH: <mi>N</mi> :MATH]
   , directly corresponds to gene count, a higher gene count necessitates
   larger dimensions. The latent dimension
   [MATH: <mtext>latent\_dim</mtext> :MATH]
   is consistently set equal to
   [MATH: <mtext>img\_size</mtext> :MATH]
   to preserve spatial coherence. Cell-type labels (
   [MATH: <mi mathvariant="bold">y</mi> :MATH]
   ) and the
   [MATH: <msub><mi>n</mi><mrow><mi
   mathvariant="italic">cls</mi></mrow></msub> :MATH]
   are adjusted simultaneously to ensure biologically relevant category
   assignments and effective imputation. In addition, we also present the
   parameter settings and convolutional layer configurations of the
   generator and discriminator.

   Generator parameters and convolutional architecture:
    1. Perform linear transformation and reshape the latent noise vector
       [MATH: <mi>z</mi> :MATH]
       through a fully connected layer, generating an initial tensor
       [MATH: <msub><mi>z</mi><mi>n</mi></msub> :MATH]
       of dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>32</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi
       >N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .
    2. Conduct convolution on
       [MATH: <msub><mi>z</mi><mi>n</mi></msub> :MATH]
       using convolutional layer
       [MATH:
       <mrow><mi>G</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><mn>1</mn><
       /mrow> :MATH]
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , obtaining tensor
       [MATH: <msub><mi>z</mi><mrow><mi
       mathvariant="italic">conv</mi></mrow></msub> :MATH]
       of dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>32</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi
       >N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .
    3. Apply the self-attention mechanism on
       [MATH: <msub><mi>z</mi><mrow><mi
       mathvariant="italic">conv</mi></mrow></msub> :MATH]
       to derive an attention-refined tensor.
    4. Perform convolution on one-hot encoded cell-type label tensors
       using convolutional layer
       [MATH:
       <mrow><mi>L</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><mn>1</mn><
       /mrow> :MATH]
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , producing tensor
       [MATH: <msub><mi>L</mi><mi>n</mi></msub> :MATH]
       of dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>8</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi>
       N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .
    5. Concatenate the attention-refined
       [MATH: <msub><mi>z</mi><mrow><mi
       mathvariant="italic">conv</mi></mrow></msub> :MATH]
       and
       [MATH: <msub><mi>L</mi><mi>n</mi></msub> :MATH]
       into tensor
       [MATH: <mrow><mi mathvariant="italic">GConcat</mi></mrow> :MATH]
       with dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>40</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi
       >N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .
    6. Execute sequential convolutional operations
       [MATH:
       <mrow><mi>G</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><msub><mn>2
       </mn><mn>1</mn></msub></mrow> :MATH]
       and
       [MATH:
       <mrow><mi>G</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><msub><mn>2
       </mn><mn>2</mn></msub></mrow> :MATH]
       , each with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , on
       [MATH: <mrow><mi mathvariant="italic">GConcat</mi></mrow> :MATH]
       , yielding the generator output tensor of dimensions
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi>
       N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .

   Discriminator parameters and convolutional architecture:
    1. Perform initial linear transformation and reshape the input tensor
       into dimensions
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>N</mi><mo>,</mo><mi>
       N</mi><mo stretchy="false">)</mo></mrow> :MATH]
       .
    2. Apply convolutional layer
       [MATH:
       <mrow><mi>D</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><mn>1</mn><
       /mrow> :MATH]
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , followed by max-pooling and a residual block, reducing the tensor
       dimension to
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>32</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow>
       :MATH]
       .
    3. Conduct convolutional layer
       [MATH:
       <mrow><mi>D</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><mn>2</mn><
       /mrow> :MATH]
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , further reducing tensor dimension to
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>16</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow>
       :MATH]
       .
    4. Integrate the self-attention mechanism on this reduced tensor to
       capture inter-feature dependencies.
    5. Perform convolution on one-hot encoded cell-type label tensors
       using convolutional layer
       [MATH:
       <mrow><mi>L</mi><mi>C</mi><mi>o</mi><mi>n</mi><mi>v</mi><mn>2</mn><
       /mrow> :MATH]
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       , generating tensor
       [MATH: <msub><mi>L</mi><mi>d</mi></msub> :MATH]
       of dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>8</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow>
       :MATH]
       .
    6. Concatenate attention-refined tensor and
       [MATH: <msub><mi>L</mi><mi>d</mi></msub> :MATH]
       forming tensor
       [MATH: <mrow><mi mathvariant="italic">DConcat</mi></mrow> :MATH]
       with dimension
       [MATH: <mrow><mo
       stretchy="false">(</mo><mn>24</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo>,</mo><mi>N</mi><mo
       stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow>
       :MATH]
       .
    7. Apply fully connected layers and additional convolutional layers
       with kernel size
       [MATH: <mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow> :MATH]
       to produce the discriminator output.

   The size of the convolutional kernel and the dimensional settings are
   adjusted based on the number of genes. The dimension typically tuned
   between 32 and 64, which is sufficient to capture the characteristics
   of various types of data. The residual parameter
   [MATH: <mi>γ</mi> :MATH]
   serves as a balancing factor within residual blocks, moderating the
   interplay between real and generated features to stabilize model
   training. Typically,
   [MATH: <mi>γ</mi> :MATH]
   is adjusted experimentally, higher values emphasize generated features,
   whereas lower values prioritize real features. This parameter is
   fine-tuned iteratively based on convergence behavior.

Metrics evaluation

   The primary objective of data imputation is to construct a “gold
   standard” dataset by rectifying false zeros introduced by technical
   noise, thereby accurately recovering the true expression levels of
   genes. In this process, genes that are genuinely unexpressed remain as
   true zeros. To mitigate the impact of zero values on computational
   analyses, minimal non-zero values are assigned to lowly expressed
   genes, while the expression levels of highly expressed genes remain
   largely unaffected.

   To rigorously assess the performance of scMASKGAN, we conducted
   comprehensive comparisons with 12 state-of-the-art imputation
   algorithms-namely, AutoImpute, DCA, DeepImpute, DrImpute, ENHANCE,
   MAGIC, SAVER, scImpute, SCRABBLE, VIPER, scIGANS, and scGAIN-across
   seven real scRNA-seq datasets. Evaluation metrics included the
   following:
     * Uniform Manifold Approximation and Projection (UMAP) distribution
       plots: Used to visualize the global structure of the data before
       and after imputation, ensuring that the overall clustering patterns
       remain biologically meaningful.
     * Z-score standardized distribution: Examined to verify that imputed
       data retains a normalized distribution, facilitating comparability
       with the original data and minimizing artificial distortions.
     * Coefficient of Variation (CV): Assessed to maintain gene expression
       variability, ensuring that important biological signals are not
       lost during imputation.
     * Wasserstein distance and Jensen-Shannon distance: Employed to
       quantify distributional differences between the imputed and
       original datasets, providing robust statistical metrics for
       evaluating imputation performance.
     * Accuracy (ACC): Evaluated to measure the ability of imputation
       methods to correctly classify cell types based on reconstructed
       expression profiles.
     * Area Under the Receiver Operating Characteristic Curve (AUC): Used
       to assess the sensitivity and specificity of gene expression
       restoration, providing a robust measure of imputation reliability.
     * F1 scores: Computed to balance precision and recall, particularly
       in identifying dropout events versus true biological zeros.
     * Pearson correlation coefficients: Calculated relative to the
       original datasets to quantify the fidelity of imputed data,
       ensuring that gene-gene relationships remain intact.

   These metrics were rigorously analyzed to provide a comprehensive
   assessment of the efficacy of scMASKGAN in accurately imputing gene
   expression data.

UMAP distribution

   UMAP is widely used for dimensionality reduction in scRNA-seq, enabling
   the visualization of high-dimensional data in two dimensions and
   facilitating data distribution analysis [[108]34]. As illustrated in
   Figure [109]2, UMAP projections of imputed Humanbrain datasets reveal
   that methods such as MAGIC, AutoImpute, DeepImpute, VIPER, scGAIN, and
   scMASKGAN effectively recover data structure. Notably, AutoImpute
   exhibits superior cell type separation, clearly distinguishing neurons,
   astrocytes, and other cell types with clustering patterns that closely
   resemble the original data. scMASKGAN performs similarly well,
   maintaining strong alignment with the inherent data structure and
   suggesting minimal alteration of the underlying biological
   distribution. While DeepImpute and VIPER also achieve reasonable cell
   type separation, minor overlaps between clusters are observed. In
   contrast, DCA, scrImpute, and scGAIN yield substantial cell type
   overlap and weaker clustering, reducing the interpretability of the
   imputed data. Overall, AutoImpute and scMASKGAN emerge as the most
   optimal imputation methods, effectively preserving cell type structure
   and distribution for robust downstream analysis.

Fig. 2.

   [110]Fig. 2
   [111]Open in a new tab

   The UMAP projections of the original scRNA-seq data alongside those of
   data imputed by 13 different methods, accompanied by cell-type labels

   Furthermore, we have presented the distributions of other datasets in
   Supplementary Figures 1 to 8. Supplementary Figures 1 and 2 display the
   UMAP distributions of the original neuroblastoma data across 10 groups
   and the corresponding imputed data generated by scMASKGAN,
   demonstrating that even under extremely high dropout rates, scMASKGAN
   maintains excellent imputation performance. Additionally, Supplementary
   Figures 3 to 8 illustrate the T-SNE and UMAP distributions of the
   datasets in Table 1, along with the imputed data distributions from
   scMASKGAN, clearly indicating that favorable imputation results are
   consistently achieved under various dropout rates.

Coefficient of variation

   The Coefficient of Variation (CV) [[112]35] serves as a standardized
   metric for quantifying data dispersion, higher CV values denote
   increased variability, whereas lower CV values indicate greater
   consistency. As illustrated in Fig. [113]3A, the CV values across
   datasets with varying dropout rates are presented. Notably, the
   SCRABBLE method exhibits comparatively elevated CV values, indicative
   of pronounced variability. In contrast, the DCA method demonstrates
   exceptionally high variability in the Mouse ESC dataset (70
   [MATH: <mo>%</mo> :MATH]
   dropout), which we attribute to potential inaccuracies in parameter
   estimation due to the large data volume. Conversely, the AutoImpute,
   scMASKGAN, and scGANs methods yield relatively lower CV values,
   reflecting superior imputation performance.

Fig. 3.

   [114]Fig. 3
   [115]Open in a new tab

   Comparison of imputation metrics. Panel A shows the CV coefficients of
   different imputation methods across various dropout rates. Panel B
   presents the Z-score standardized distribution comparison between
   scMASKGAN imputed data and the original data under extreme missingness.
   Panel C illustrates the CV coefficients comparison between scMASKGAN
   imputed data and the original data under extreme missingness
   conditions. Panel D compares multiple imputation methods using the JS
   test and Wasserstein distance

   Furthermore, to rigorously assess the robustness of scMASKGAN under
   extreme missing data conditions, Fig. [116]3C displays the CV values
   for scMASKGAN’s imputed results across 10 neuroblastoma dataset groups.
   In conjunction with the UMAP distribution analyses, these findings
   suggest that scMASKGAN effectively integrates low variability with high
   fidelity to the original data distribution, thereby representing an
   exemplary imputation approach that is critical for the accuracy of
   downstream analyses.

Z-score distribution

   Z-score [[117]36] distribution refers to the distribution obtained by
   applying Z-score normalization to the data. Specifically, for each data
   point
   [MATH: <mi>x</mi> :MATH]
   , the Z-score is defined as
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><mi>z</mi><mo>=</mo><mfrac><mrow><mi>x</mi><m
   o>-</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac></mrow></mtd></mtr></mtable>
   </mrow> :MATH]
   7

   where
   [MATH: <mi>μ</mi> :MATH]
   is the mean of the data and
   [MATH: <mi>σ</mi> :MATH]
   is the standard deviation. This normalization procedure transforms the
   data such that the resulting distribution has a mean of 0 and a
   standard deviation of 1. We utilize the Z-score distribution to compare
   the original data with the imputed data and to detect significant
   deviations. A more concentrated Z-score distribution indicates fewer
   outliers and superior data quality. As shown in Supplementary Figure 9,
   the Z-score distributions for datasets with varying dropout rates
   demonstrate that scMASKGAN performs outstandingly. Moreover,
   Fig. [118]3B presents the Z-score distributions of both the original
   and the scMASKGAN-imputed data under extremely high dropout rates. When
   combined with the UMAP distribution and CV coefficient analyses from
   the preceding sections, these results indicate that scMASKGAN maintains
   excellent performance even under extreme conditions.

Statistical tests

   In Fig. [119]3D, we compare the JS distance [[120]37] and the
   Wasserstein distance (EMD) [[121]38] between the imputed data generated
   by various methods and the original data. The JS distance is computed
   derived from the Jensen-Shannon divergence, which is defined as
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><mtext>JSD</mtext><mrow><mo
   stretchy="false">(</mo><mi>P</mi><mo
   stretchy="false">‖</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mf
   rac><mspace
   width="0.166667em"></mspace><msub><mi>D</mi><mtext>KL</mtext></msub><mf
   enced close=")" open="("><mi>P</mi><mspace
   width="0.166667em"></mspace><mrow><mo maxsize="2.047em"
   minsize="2.047em" stretchy="true">‖</mo></mrow><mspace
   width="0.166667em"></mspace><mfrac><mrow><mi>P</mi><mo>+</mo><mi>Q</mi>
   </mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn
   ></mfrac><mspace
   width="0.166667em"></mspace><msub><mi>D</mi><mtext>KL</mtext></msub><mf
   enced close=")" open="("><mi>Q</mi><mspace
   width="0.166667em"></mspace><mrow><mo maxsize="2.047em"
   minsize="2.047em" stretchy="true">‖</mo></mrow><mspace
   width="0.166667em"></mspace><mfrac><mrow><mi>P</mi><mo>+</mo><mi>Q</mi>
   </mrow><mn>2</mn></mfrac></mfenced></mrow></mtd></mtr></mtable></mrow>
   :MATH]
   8

   where
   [MATH: <msub><mi>D</mi><mtext>KL</mtext></msub> :MATH]
   denotes the Kullback–Leibler divergence. The JS distance is then
   obtained as the square root of the Jensen-Shannon divergence:
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><mtext>JSdistance</mtext><mrow><mo
   stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><msqrt><mrow><mtext>JSD</mtext>
   <mo stretchy="false">(</mo><mi>P</mi><mo
   stretchy="false">‖</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow></msqrt></mrow></mtd></mtr></mtable></mro
   w> :MATH]
   9

   The Wasserstein distance (also known as the Earth Mover’s Distance,
   EMD) is defined as
   [MATH: <mrow><mtable><mtr><mtd
   columnalign="right"><mrow><msub><mi>W</mi><mn>1</mn></msub><mrow><mo
   stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow><mo>=</mo><munder><mo
   movablelimits="true">inf</mo><mrow><mi>γ</mi><mo>∈</mo><mi
   mathvariant="normal">Γ</mi><mo
   stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow></munder><msub><mo>∫</mo><mrow><mi
   mathvariant="script">X</mi><mo>×</mo><mi
   mathvariant="script">X</mi></mrow></msub><mrow><mo
   stretchy="false">‖</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo
   stretchy="false">‖</mo></mrow><mspace
   width="0.166667em"></mspace><mi>d</mi><mi>γ</mi><mrow><mo
   stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo
   stretchy="false">)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></m
   row> :MATH]
   10

   where
   [MATH: <mrow><mi mathvariant="normal">Γ</mi><mo
   stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo
   stretchy="false">)</mo></mrow> :MATH]
   denotes the set of joint distributions with marginals
   [MATH: <mi>P</mi> :MATH]
   and
   [MATH: <mi>Q</mi> :MATH]
   , and
   [MATH: <mrow><mo
   stretchy="false">‖</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo
   stretchy="false">‖</mo></mrow> :MATH]
   represents the distance between
   [MATH: <mi>x</mi> :MATH]
   and
   [MATH: <mi>y</mi> :MATH]
   in the space
   [MATH: <mi mathvariant="script">X</mi> :MATH]
   .

   Our results indicate that scMASKGAN exhibits outstanding performance
   with respect to both metrics, as evidenced by its low JS distance and
   low Wasserstein distance, which suggest a high degree of consistency
   between the imputed data and the original data. In contrast, the
   AutoImpute, scGAIN, sciGANs, and SCRABBLE methods show only moderate
   performance in terms of the Wasserstein distance. Furthermore, sciGANs
   and AutoImpute perform poorly in terms of the JS distance. A closer
   examination of the imputed data reveals that sciGANs tends to impute
   all missing values, which does not conform to the requirement of
   removing dropout values and consequently introduces a large number of
   spurious biological signals. Additionally, in order to achieve higher
   metric scores, AutoImpute introduces negative values into the gene
   expression matrix, a practice that is biologically implausible.

   This observation highlights the limitations in adaptability and
   stability of both scImpute and Deepimpute across diverse datasets. In
   particular, while scImpute can effectively impute data from certain
   platforms, its output for other datasets results in missing values when
   calculating the JS distance, EMD statistics, and Z-score distributions,
   rendering these metrics uncomputable. Meanwhile, DeepImpute produced
   usable results solely in the Human Brain dataset. These findings
   suggest that some methods may require further optimization or
   integration with other approaches to enhance estimation performance and
   ensure the reliability of subsequent metric calculations.

Cluster metrics

   ACC, AUC, and F1 score are standard classification evaluation metrics
   used in clustering, with values ranging from 0 to 1, where higher
   values indicate better clustering performance[[122]39]. To further
   assess the effectiveness of various imputation algorithms, we employed
   the Louvain algorithm to cluster seven datasets and compared the
   resulting clustering labels with the corresponding cell types. The
   performance in terms of ACC, AUC, and F1 scores is summarized in Tables
   [123]3 and [124]4. Table [125]3 details different types of scRNA-seq
   data, with ”...” indicating that certain algorithms were not applicable
   to specific datasets, while Table [126]4 records scRNA-seq data from
   different platforms. Figure [127]4 visualizes these results.

Table 3.

   Clustering metrics for 13 imputation algorithms across various datasets
   Datasets Human brain ERCC spike-in Mouse ESC Time-course
   Metric ACC AUC F1 ACC AUC F1 ACC AUC F1 ACC AUC F1
   AutoImpute 0.876 0.745 0.634 0.608 0.578 0.453 0.518 0.575 0.518 0.743
   0.669 0.507
   DCA 0.781 0.669 0.472 0.542 0.538 0.434 0.502 0.666 0.506 0.500 0.667
   0.504
   Deepimpute 0.803 0.678 0.492 0.559 0.557 0.454 0.640 0.730 0.721 0.846
   0.865 0.874
   DrImpute 0.834 0.732 0.580 0.604 0.61 0.513 ... ... ... ... ... ...
   ENHANCE 0.862 0.751 0.626 0.746 0.718 0.624 ... ... ... ... ... ...
   MAGIC 0.812 0.689 0.512 0.563 0.542 0.423 0.515 0.670 0.515 0.500 0.667
   0.494
   SAVER 0.856 0.804 0.671 0.525 0.526 0.425 ... ... ... ... ... ...
   scGAIN 0.626 0.562 0.332 0.359 0.501 0.491 0.749 0.772 0.808 0.739
   0.792 0.789
   scImpute 0.857 0.767 0.638 0.711 0.734 0.649 0.904 0.910 0.966 0.977
   0.977 0.998
   SCRABBLE 0.481 0.575 0.367 0.540 0.522 0.404 ... ... ... ... ... ...
   VIPER 0.768 0.662 0.460 0.511 0.507 0.404 ... ... ... ... ... ...
   scIGANs 0.802 0.678 0.492 0.546 0.525 0.405 0.812 0.785 0.940 0.542
   0.684 0.549
   scMASKGAN 0.975 0.965 0.964 0.54 0.954 0.667 0.614 0.704 0.684 0.946
   0.948 0.949
   [128]Open in a new tab

   The bold indicates the best results of the test metrics across
   different datasets

Table 4.

   Clustering metrics for 13 imputation algorithms across platform
   datasets
   Datasets   sc_10X            sc_CELseq2        sc_Dropseq
   Metric     ACC   AUC   F1    ACC   AUC   F1    ACC   AUC   F1
   AutoImpute 0.979 0.977 0.969 0.577 0.565 0.461 0.821 0.800 0.737
   DCA        0.997 0.977 0.997 0.995 0.994 0.993 0.935 0.926 0.905
   Deepimpute 0.997 0.997 0.996 0.990 0.988 0.985 0.994 0.993 0.991
   DrImpute   0.996 0.995 0.993 0.980 0.977 0.970 0.909 0.897 0.865
   ENHANCE    0.791 0.800 0.726 0.958 0.976 0.996 0.994 0.993 0.991
   MAGIC      0.953 0.948 0.930 0.546 0.507 0.366 0.546 0.535 0.429
   SAVER      0.996 0.995 0.993 0.990 0.988 0.985 0.994 0.993 0.991
   scGAIN     0.880 0.872 0.825 0.500 0.508 0.424 0.481 0.500 0.425
   scImpute   0.658 0.668 0.503 0.940 0.883 0.787 0.930 0.910 0.890
   SCRABBLE   0.990 0.988 0.985 0.501 0.500 0.406 0.519 0.511 0.409
   VIPER      0.999 0.998 0.999 0.564 0.517 0.365 0.923 0.920 0.890
   scIGANs    0.658 0.668 0.503 0.430 0.035 0.601 0.412 0.667 0.500
   scMASKGAN  0.999 0.981 0.966 0.969 0.503 0.668 0.997 0.836 0.803
   [129]Open in a new tab

   The bold indicates the best results of the test metrics across
   different datasets

Fig. 4.

   Fig. 4
   [130]Open in a new tab

   The clustering metrics for imputation algorithms compared to cell-type
   labels. The figure presents boxplots comparing ACC, AUC, and F1 scores
   of 13 imputation methods across different datasets, with each color
   representing a different method

   scMASKGAN outperforms other methods in terms of ACC, followed by DCA,
   DeepImpute, and SAVER, which also show strong performance from the
   analysis. In terms of the F1 score, scMASKGAN, DeepImpute, and SAVER
   exhibit the highest median values. For AUC, scMASKGAN, DeepImpute, and
   SAVER again show leading performance, with scMASKGAN maintaining a high
   median and demonstrating the most stable distribution, as indicated by
   the boxplot. These results suggest that scMASKGAN not only excels
   across all metrics but also demonstrates superior stability and
   generalizability, making it well-suited for diverse scRNA-seq data
   types, sizes, and platforms.

Correlation analysis

   In addition to comparing clustering performance, we evaluated each
   imputation method by analyzing the Pearson correlation between the
   imputed and original data [[131]40]. Figure [132]5 presents boxplots of
   the Pearson correlation coefficients, where scMASKGAN, SAVER, scImpute,
   and SCRABBLE show higher median values, indicating a strong alignment
   with the original data and, thus, more reliable imputation. In
   contrast, MAGIC, ENHANCE, and VIPER display lower correlation values,
   implying that their imputed data deviate more from the observed
   values-likely due to the introduction of bias during imputation.

Fig. 5.

   Fig. 5
   [133]Open in a new tab

   Boxplot comparison of Pearson correlation coefficients among 13
   imputation methods

   In conclusion, integrating both clustering performance and correlation
   analyses, scMASKGAN and SAVER emerge as the most optimal imputation
   methods, excelling in preserving data consistency and stability.
   Conversely, ENHANCE and MAGIC show limitations in both aspects, making
   them less suitable for tasks that require high-fidelity imputation.

Cost analysis

   To rigorously assess the performance of scMASKGAN, we conducted
   benchmarking experiments using a computational platform comprising two
   NVIDIA 4090 GPUs (24GB each) and one NVIDIA A4000 GPU (16GB). We
   evaluated both the memory consumption and the runtime efficiency of 13
   imputation methods, including scMASKGAN, on gene expression datasets
   spanning 1,000 to 100,000 cells. As illustrated in Fig. [134]6,
   scMASKGAN demonstrates competitive performance, achieves second place
   in memory consumption-surpassed only by scIGAN-while its execution
   speed is slightly lower than that of the MAGIC and DeepImpute methods.
   Overall, these findings underscore the robust computational performance
   of scMASKGAN, affirming its efficacy in terms of both memory usage and
   runtime efficiency for large-scale single-cell transcriptomic analyses.

Fig. 6.

   Fig. 6
   [135]Open in a new tab

   Comparison of the memory consumption and the runtime efficiency among
   13 imputation methods

Downstream analysis

   We demonstrated the biological relevance of scMASKGAN for data recovery
   through comprehensive validation using several biological datasets.
   Specifically, we utilized Mouse ESC scRNA-seq data, temporal data, and
   10 neuroblastoma sample datasets. We conducted a series of analyses,
   including gene-gene correlation analysis, temporal analysis, gene
   enrichment pathway analysis, batch data imputation, and differential
   gene expression analysis, to assess the performance of scMASKGAN across
   different types of biological data. These analyses not only help
   clarify the role of scMASKGAN in biological data recovery but also
   provide strong evidence for its potential use in a variety of
   biological research applications.

Gene-gene correlation

   Gene-gene correlation reveals similarities in expression patterns,
   aiding in the identification of regulatory networks and gene functions
   [[136]41]. We analyzed this using a Mouse ESC scRNA-seq dataset,
   previously applied in scIGAN studies, which comprises 44 cell cycle
   genes across over 6,800 cells. Figure [137]7A shows confusion matrices
   based on Pearson correlation coefficients for both original and imputed
   data. Notably, significant correlations-such as those among Mcm6, Nasp,
   Mcm2, and Pcna, as well as between Cdk1, Cenpl, and Top2a-are preserved
   post-imputation. Additionally, new correlations between cdc20 and
   Cenpa/PLK1, and between Msh2 and Mcm2/Mcm6, align with known
   co-expression relationships [[138]42, [139]43]. Hierarchical clustering
   further confirms that the imputed data retains the original data’s
   structural characteristics. Overall, these results validate that
   scMASKGAN effectively restores biologically meaningful gene
   correlations while preserving the inherent data structure.

Fig. 7.

   [140]Fig. 7
   [141]Open in a new tab

   Gene-gene correlation analysis for mouse ESC data, trajectory analysis
   of Time-course scRNA-seq data, and gene pathway enrichment analysis.
   Panel A shows a heatmap of gene-gene correlations for scMASKGAN-imputed
   versus original data, with the left color bar representing hierarchical
   clustering results and the middle section showing a confusion matrix of
   44 gene groups where darker blue indicates stronger correlations. Panel
   B displays trajectory analysis plots for original and imputed data.
   Panel C shows temporal profiles for six marker genes, with the red line
   indicating imputed data and the blue line showing original data. Panel
   D features bar and bubble charts of GO enrichment analysis for FCER1A,
   used to assess the reasons for its temporal upregulation

Temporal analysis

   Temporal analysis is employed in scRNA-seq to infer cellular
   trajectories during dynamic processes [[142]44]. In this study, we
   imputed a time-course scRNA-seq dataset-capturing the differentiation
   of H1 embryonic stem cells (ESCs) into definitive endodermal cells
   (DEC)-using scMASKGAN, and then reconstructed trajectories with the
   Monocle3 R package [[143]45]. As illustrated in Fig. [144]7B, UMAP
   plots of the original data reveal distinct, dispersed clusters due to
   technical noise, while the scMASKGAN-imputed data show smoother
   transitions and enhanced connectivity between cells. Further temporal
   analysis of marker genes (Fig. [145]7C) indicates that genes such as
   CD14, CD3D, MS4A1, and PPBP exhibit increased expression along clearer
   trajectories, whereas FCER1A, previously affected by dropout events,
   demonstrates a gradual expression increase over time. GO enrichment
   analysis (Fig. [146]7D) confirms that these genes are primarily
   associated with immune functions, and suggesting that the progressive
   activation of immune-related pathways underlies the differentiation
   process from ESC to DEC.

Batch data imputation

   We use scRNA-seq data from the same batch to impute and integrate 10
   homologous neuroblastoma samples from clinical cases. All data undergo
   a complete Scanpy [[147]46] quality control pipeline: ensuring that
   each gene is expressed in at least 3 cells, each cell contains at least
   200 genes, and cells with mitochondrial gene expression exceeding 5
   [MATH: <mo>%</mo> :MATH]
   are removed. Subsequently, the data are log-transformed and subjected
   to PCA for dimensionality reduction. These preprocessing steps enable
   us to assess whether the original gene expression patterns are
   maintained and to evaluate the restoration of key marker genes.

   As shown in Fig. [148]8A, we present UMAP plots of the original data
   alongside datasets imputed by scMASKGAN, DCA, and DeepImpute. The
   results indicate that scMASKGAN enhances cell connectivity and
   effectively reduces the technical noise present in the original data.
   In contrast, although the DCA-imputed data exhibit a more dispersed
   clustering that still delineates cell population differences, the
   excessive fragmentation may compromise biological relevance. Thus,
   following batch effect correction and integration, scMASKGAN better
   reflects the underlying cellular states and connectivity, offering
   superior imputation performance.

Fig. 8.

   [149]Fig. 8
   [150]Open in a new tab

   The results of batch data imputation and differential gene expression
   analysis are illustrated in three panels. Panel A displays UMAP plots
   comparing batch-integrated data from three imputation
   algorithms-scMASKGAN, DCA, and Deepimpute-to the original dataset.
   Panel B shows the expression levels of five marker genes and the
   interference gene (NTRK2) in both the original and scMASKGAN-imputed
   data, with red indicating original data and blue indicating imputed
   data. Panel C presents scatter plots of the corresponding gene
   expressions, with the color bar on the right representing gene
   expression levels

   Additionally, we selected one low-risk neuroblastoma ([151]GSM5768752)
   sample to compare the expression of five marker genes and a control
   gene (NTRK2, which is highly expressed in high-risk neuroblastoma)
   [[152]47] between the scMASKGAN-imputed and original datasets. As shown
   in supplementary Fig. [153]8B, scMASKGAN-imputed data exhibit minimal
   changes in high-expression genes while notably restoring low-expression
   genes. Furthermore, supplementary Fig. [154]8C demonstrates that,
   except for NTRK2, the expression levels of genes with initially low
   expression increase, whereas high-expression genes remain largely
   unchanged. These observations confirm that scMASKGAN achieves high
   accuracy in data recovery without indiscriminately altering gene
   expression.

Gene expression analysis

   To further validate the effectiveness of the scMASKGAN imputation, we
   compared the expression profiles of commonly highly variable genes
   across different cell types between the imputed and original data under
   various dropout rates, as illustrated in Fig. [155]9. Specifically, we
   present the comparison for the Human brain dataset (81.4
   [MATH: <mo>%</mo> :MATH]
   dropout) and observe that the data structure remains intact, with no
   significant alterations in the expression of highly variable genes. In
   addition, we analyzed datasets with other dropout rates in
   Supplementary Figure 10, where scMASKGAN consistently preserved the
   complete biological signal. These results further demonstrate the
   accuracy and reliability of the scMASKGAN imputation.

Fig. 9.

   [156]Fig. 9
   [157]Open in a new tab

   Comparison of common highly variable genes between imputed data and
   original data across different cell types

Discussion

Robustness across diverse datasets and platforms

   To comprehensively evaluate the strengths and weaknesses of scMASKGAN,
   we compared it with 12 existing imputation methods across various
   scRNA-seq datasets. In terms of the CV, scMASKGAN’s performance is
   surpassed only by AutoImpute and scIGANs (Fig. [158]3A). For EMD and JS
   distance metrics (Fig. [159]3D), while most methods exhibit robust
   performance, scImpute and DeepImpute were excluded from some
   comparisons due to their inability to impute certain datasets, the
   suboptimal performance of sciGAN and AutoImpute relative to other
   methods has been discussed in Sect. [160]3.4. Moreover, clustering
   assessments (as detailed in Tables [161]3 and [162]4, and Fig. [163]4)
   indicate that scMASKGAN, alongside SAVER, yields the most effective
   segregation of cellular heterogeneity. Pearson correlation analysis
   further confirms that scMASKGAN’s imputed data maintains the highest
   correlation with the original data, and the Z-score standardized
   distribution (Supplementary Figure 9) demonstrates that its imputation
   results exhibit minimal outliers and remarkable stability across
   varying dropout rates. Collectively, these findings underscore the
   superior efficacy of scMASKGAN as an imputation method.

   The results also indicate that scMASKGAN performed exceptionally well
   on the ERCC spike-in dataset, Human brain dataset, and Time-course
   dataset, demonstrating its advantages in handling highly sparse,
   small-scale, and temporal sequencing data. This superior performance
   may be attributed to the adversarial training mechanism, which enables
   scMASKGAN to effectively learn the latent data distribution and
   generate imputed values that closely resemble the true expression
   patterns. For the Human brain dataset, scMASKGAN successfully recovered
   gene expression patterns while preserving cellular heterogeneity. In
   the Time-course dataset, it maintained the dynamic characteristics
   inherent in temporal sequencing data, thereby avoiding the loss of
   time-related information often encountered in traditional imputation
   methods.

   On the large-scale Mouse ESC dataset, scMASKGAN demonstrated moderate
   performance, without consistently outperforming all comparative
   methods. As evidenced by Supplementary Figure 4, although scMASKGAN
   effectively delineated three distinct clusters, the corresponding
   clustering metrics remained only moderate. This may be attributable to
   alterations in the UMAP embedding distribution following cluster
   separation, potentially introducing inaccuracies in the clustering
   indices. Nevertheless, with respect to other evaluation criteria,
   scMASKGAN consistently produced biologically meaningful imputed values
   and preserved the overall structural integrity of the dataset.

   For the sc_10X and sc_Drop-seq datasets, scMASKGAN exhibited robust
   performance, underscoring its strong generalization capabilities in
   high-throughput sequencing scenarios. These datasets, characterized by
   substantial technical noise and sparsity, benefit from scMASKGAN’s
   integrated attention mechanism and adversarial training framework,
   which collectively enable the effective discrimination of biological
   signals from technical artifacts, thereby yielding highly accurate
   imputed values. Moreover, in the neuroblastoma dataset with
   exceptionally high dropout rates, scMASKGAN demonstrated outstanding
   imputation performance, further confirming its capacity to enhance data
   quality under extreme conditions.

   The performance on the sc_CEL-seq2 dataset was suboptimal, likely due
   to its unique characteristics and high levels of sequencing noise. Our
   imputation results did not fully recover the expression profiles of
   several key genes, indicating that scMASKGAN could benefit from further
   refinement to better accommodate various sequencing platforms.
   Moreover, the pronounced technical noise in sc_CEL-seq2 appears to
   destabilize the adversarial training process of the GAN-based model,
   resulting in incomplete gene expression recovery.

Enhanced interpretability and downstream analysis

   The downstream analyses presented in our study provide comprehensive
   evidence of the efficacy and robustness of the scMASKGAN imputation
   method. The gene-gene correlation analysis demonstrates that scMASKGAN
   accurately restores biologically meaningful relationships between key
   cell cycle genes, such as the significant correlations observed among
   Mcm6, Nasp, Mcm2, and Pcna, as well as between Cdk1, Cenpl, and Top2a.
   Moreover, the emergence of novel correlations that align with known
   co-expression patterns further substantiates the ability of scMASKGAN
   to recover latent gene-gene interactions while preserving the intrinsic
   data structure.

   The temporal analysis reinforces these findings by illustrating
   scMASKGAN’s capacity to capture dynamic cellular trajectories. As
   evidenced by the UMAP plots in Fig. [164]7B, the imputed data exhibit
   smoother transitions and enhanced connectivity compared to the original
   noisy data, thereby facilitating the reconstruction of continuous
   cellular differentiation pathways. Additionally, the temporal
   expression profiles of marker genes reveal that dropout-affected genes,
   such as FCER1A, regain a gradual expression increase over time, which
   is critical for deciphering time-related biological processes. The
   corresponding GO enrichment analysis further confirms that these genes
   are predominantly involved in immune functions, underscoring the
   biological relevance of the imputation.

   In the context of batch data imputation (Sect. [165]4.3 and Fig.
   [166]8), scMASKGAN not only attenuates technical noise but also
   effectively integrates data from multiple samples. Compared to other
   methods, UMAP visualizations (Fig. [167]8A) reveal that most algorithms
   lead to significant fragmentation in gene expression profiles as a
   result of imputation. Furthermore, differential expression analyses of
   key marker genes (Fig. [168]8B and C) reveal that scMASKGAN not only
   preserves the expression profiles of highly expressed genes and
   accurately recovers missing signals, but also avoids introducing
   erroneous biological artifacts, thereby maintaining the overall
   integrity of the biological signal.

   Finally, the gene expression analysis across datasets with varying
   dropout rates (Fig. [169]9 and Supplementary Figure 10) further
   corroborates the robustness of scMASKGAN. The imputed data retain the
   expression profiles of highly variable genes, suggesting that the
   method effectively preserves critical biological signals without
   introducing significant distortions.

Conclusion

   In summary, these results collectively demonstrate that scMASKGAN is a
   robust and versatile imputation method. It effectively recovers both
   global and local gene expression patterns across diverse scRNA-seq
   datasets, ranging from small-scale and highly sparse data to large
   heterogeneous datasets and time-course studies. While certain
   platform-specific limitations remain, the overall performance of
   scMASKGAN underscores its potential as a powerful tool for improving
   data quality in downstream single-cell analyses. Future research will
   focus on further optimizing the adversarial training strategy and
   tailoring the model to the specific characteristics of different
   sequencing platforms.

Supplementary Information

   [170]Supplementary material 1^ (2.3MB, png)
   [171]Supplementary material 2^ (2.4MB, png)
   [172]Supplementary material 3^ (70.7KB, pdf)
   [173]Supplementary material 4^ (1.4MB, pdf)
   [174]Supplementary material 5^ (58.4KB, pdf)
   [175]Supplementary material 6^ (69.8KB, pdf)
   [176]Supplementary material 7^ (210.5KB, pdf)
   [177]Supplementary material 8^ (174.3KB, pdf)
   [178]Supplementary material 9^ (1.3MB, pdf)
   [179]Supplementary material 10^ (201.9KB, pdf)
   [180]Supplementary material 11^ (620B, csv)
   [181]Supplementary material 12^ (16.8KB, xlsx)

Acknowledgements