Title: Gene-Gene Relationship Modeling Based on Genetic Evidence for Single-Cell RNA-Seq Data Imputation
Venue: NeurIPS 2024
Accepted: yes
Keywords: scRNA-seq

imputation

bioinformatics
Abstract: Single-cell RNA sequencing (scRNA-seq) technologies enable the exploration of cellular heterogeneity and facilitate the construction of cell atlases. However, scRNA-seq data often contain a large portion of missing values (false zeros) or noisy values, hindering downstream analyses. To recover these false zeros, propagation-based imputation methods have been proposed using $k$-NN graphs. However they model only associating relationships among genes within a cell, while, according to well-known genetic evidence, there are both associating and dissociating relationships among genes. To apply this genetic evidence to gene-gene relationship modeling, this paper proposes a novel imputation method that newly employs dissociating relationships in addition to associating relationships. Our method constructs a $k$-NN graph to additionally model dissociating relationships via the negation of a given cell-gene matrix. Moreover, our method standardizes the value distribution (mean and variance) of each gene to have standard distributions regardless of the gene. Through extensive experiments, we demonstrate that the proposed method achieves exceptional performance gains over state-of-the-art methods in both cell clustering and gene expression recovery across six scRNA-seq datasets, validating the significance of using complete gene-gene relationships in accordance with genetic evidence. The source code is available at https://github.com/daehoum1/scCR.

Gene-Gene Relationship Modeling Based on Genetic Evidence for Single-Cell RNA-Seq Data Imputation

Abstract

Single-cell RNA sequencing (scRNA-seq) technologies enable the exploration of cellular heterogeneity and facilitate the construction of cell atlases. However, scRNA-seq data often contain a large portion of missing values (false zeros) or noisy values, hindering downstream analyses. To recover these false zeros, propagation-based imputation methods have been proposed using k -NN graphs. However they model only associating relationships among genes within a cell, while, according to well-known genetic evidence, there are both associating and dissociating relationships among genes. To apply this genetic evidence to gene-gene relationship modeling, this paper proposes a novel imputation method that newly employs dissociating relationships in addition to associating relationships. Our method constructs a k -NN graph to additionally model dissociating relationships via the negation of a given cell-gene matrix. Moreover, our method standardizes the value distribution (mean and variance) of each gene to have standard distributions regardless of the gene. Through extensive experiments, we demonstrate that the proposed method achieves exceptional performance gains over state-of-the-art methods in both cell clustering and gene expression recovery across six scRNA-seq datasets, validating the significance of using complete gene-gene relationships in accordance with genetic evidence. The source code is available at https: //github.com/daehoum1/scCR .

1 Introduction

Single-cell RNA sequencing (scRNA-seq) has become one of the most widely used technologies in biomedical research due to its ability to measure genome-wide gene expression at the single-cell level [1-3]. ScRNA-seq enables us to discover novel cell types [4], analyze cellular trajectories [5], and improve understanding human disease [6, 7]. However, scRNA-seq analysis encounters significant challenges due to the high rate of zero values in scRNA-seq data represented by a cell-gene matrix. Specifically, owing to the low RNA capture rate, scRNA-seq data often contain zero values. These zero values represent unobserved gene expression resulting from both technical omissions (referred to as dropouts [8]) and true biological absence. Moreover, even non-zero values in scRNA-seq data suffer from various sources of noise, such as cell cycle effects and batch effects [9, 10].

Ji Won Yoon

Yunha Yeo

To deal with the missing or noisy gene expression in scRNA-seq data, diverse imputation methods have been proposed, which can be categorized into non-graph-based, graph neural network (GNN)based, and propagation-based methods. Among these methods, propagation-based methods [11, 12] have been favored due to their outstanding performance. The propagation-based methods construct a k -nearest neighbor ( k -NN) graph on scRNA-seq data represented as a cell-gene matrix, and fill in missing values by propagating nonzero values on the k -NN graph. Despite their effectiveness, they overlook well-known genetic evidence [13, 14], which means that there are two types of relationships between genes: associating relationship and dissociating one. As shown in Figure 1, associating relationships represent genes that co-occur, whereas dissociating relationships represent genes that avoid co-occurrence.

However, the existing methods cannot model the dissociating gene-gene relationship by constructing a simple k -NN graph to connect only the associating genes with similar occurrence patterns. Consequently, these methods fail to connect dissociating genes. Within a cell, when considering the value to be imputed for gene Q, the value for its associating gene can assist in inferring the value for gene Q. However, its dissociating gene can also provide crucial information: if its dissociating gene has a high value, the value for gene Q may be low, as they tend to avoid each other. Additionally, the value distribution of a gene often differs significantly from that of other genes [15]. Therefore, the sum of propagated values from other genes may lead to the mixing of values at various scales, which is not suitable for data recovery.

To resolve the aforementioned problems, we propose a novel propagation-based imputation scheme called Single-Cell Complete Relationship (scCR) for scRNA-seq data, which models both associating and dissociating gene-gene relationships. scCR concatenates a given cell-gene matrix and its negation, then standardizes the value distribution (mean and variance) in each column ( i.e. , gene) of the concatenated matrix. Subsequently, we construct a k -NN graph on this concatenated and standardized matrix to connect both associating and dissociating genes within a cell. Through a propagation process on this k -NN graph, scCR effectively denoises scRNA-seq data by capturing complete genegene relationships. Extensive experimental results demonstrate that scCR significantly outperforms state-of-the-art methods in both gene expression recovery and cell clustering. Through experiment, we further confirm that scCR can model dissociating gene-gene relationships inherent in scRNA-seq data.

The main contributions of our work are summarized as follows: (1) We newly propose an effective imputation method for scRNA-seq data, which is based on the genetic evidence. Our method can model complete gene-gene relationships, including both associating and dissociating relationships; (2) We employ a standardization step before propagation among genes for additional performance improvement in downstream tasks on scRNA-seq data; (3) By modeling dissociating gene-gene relationships and utilizing the standardization step, our scCR significantly improves performance in various downstream tasks, outperforming the state-of-the-art methods by a large margin.

2 Related Work

Handling noise in scRNA-seq data. Approaches for handling noise in scRNA-seq data can be categorized into non-graph-based, GNN-based, and propagation-based methods. As pioneering efforts to impute zero values, non-graph-based methods predominantly employ either statistical techniques [16, 17] or autoencoder frameworks [17, 18]. Building on this foundation, graph-based approaches, including GNN-based and propagation-based methods, have received significant attention due to their ability to model relationships among cells and genes through graph structures. scGNN [19] leverages cell-cell relationships by constructing a cell-cell similarity matrix within a graph autoencoder framework. scGCL [20] is a graph autoencoder framework that exploits contrastive learning to capture cell-cell relationships. scTAG [21] is a clustering method that employs a graph autoencoder framework using a cell-cell k -NN graph, which jointly optimizes clustering loss and reconstruction loss.

Propagation-based imputation in scRNA-seq data. Propagation-based imputation methods have shown their superiority in scRNA-seq data imputation. They promote greater similarity in gene expression among cells that are already similar through iterative propagation steps. While MAGIC [22] utilizes a diffusion mechanism to denoise scRNA-seq data, updating values through the diffusion of both zero values and observed nonzero values may be significantly affected by false zero values ( i.e. , dropouts). To address this issue, Feature Propagation (FP) [23] can be a good solution because FP preserves observed values during diffusion while updates unobserved values through

Xcom

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Xcom

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gene-geneFPusing Agene

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Stage 2: Complete Relationship

Genes

cell-cellFP

using Acell1)

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using Acel(2)

X(2)

Acel(3)

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Acell(2)

Stage 3: Denoising

Acell(3)

diffusion. Propagation-based imputation methods have shown their superiority in scRNA-seq data imputation. scFP [11] adopts FP developed for graph-structured data to resolve imputation for scRNA-seq data. scFP constructs a cell-cell k -NN graph and applies FP for the imputation of zero values. Very recently, [12] proposes scBFP to utilize gene-gene relationships as well as cell-cell relationships. scBFP consists of two stages, and in each stage, it applies FP using a gene-gene k -NN graph and a cell-cell k -NN graph, respectively. Although scBFP is designed to leverage gene-gene relationships, the simple addition of FP using a gene-gene k -NN graph cannot effectively exploit gene-gene relationships due to the following two reasons: (1) a gene-gene k -NN graph can connect only associating genes which have co-occurrence relationships while overlooking the presence of dissociating gene-gene relationships; (2) Since the distributions for each gene significantly varies, propagation without additional processing will degrade recovery performance.

3 Preliminaries

Notation. A graph can be represented as G = ( V , E ) , where V = { v 1 , . . . , v N } is the set of N nodes and E is the set of edges. The connectivity of G can be represented by the adjacency matrix A ∈ { 0 , 1 } N × N with A i,j = 1 iff ( v i , v j ) ∈ E and A i,j = 0 otherwise. Given an arbitrary matrix B ∈ R a × b , we let k NN ( · ) : R a × b →{ R } a × a be a function that generates a normalized adjacency matrix A of the row-row k -NN graph based on cosine similarity. Here, the normalized adjacency matrix A is obtained by A = D -1 / 2 AD -1 / 2 where A is the adjacency matrix of the k -NN graph and D is a degree matrix with diagonal entries D i,i = ∑ j A i,j . Consequently, while k NN ( X ) yields A cell ∈ { 0 , 1 } C × C of the cell-cell k -NN graph from X , k NN ( X ⊤ ) produces A gene ∈ { 0 , 1 } G × G of the gene-gene k -NN graph from X . We let B i, : and B : ,j denote the i -th row vector of B and the j -th column vector of B , respectively.

Feature Propagation. FP-based algorithms [23, 24] are proposed to impute missing features in graph-structured data. The core idea of FP is to impute missing values by diffusing observed values while preserving these observed values. Assume that a given graph G = ( V , E ) has a feature matrix X ∈ R N × F with missing values, where rows and columns correspond to nodes and F feature channels, respectively. We use A ∈ R N × N to denote a normalized adjacency matrix of the graph. To preserve known features during the diffusion process, we mark the positions of the features to be

preserved with 1 in the mask matrix M ∈ { 0 , 1 } N × F ; here, values of 1 in M indicate the location of observed features. We express FP as a function by X = FP ( X , A , M ) where X ∈ R N × F is an output matrix. A detailed explanation of FP is provided in Appendix A. In summary, FP fills in missing values in X through diffusion using A while preserving features corresponding to values of 1 in M . It is noteworthy that FP ( X , A , M ) performs propagation among the rows of X . In scRNA-seq data imputation, FP-based imputation methods treat zero values as missing values to be imputed via features diffused from non-zero values.

4 Proposed Method

4.1 Overview of scCR

In this paper, we design a novel imputation framework for scRNA-seq data, namely scCR, which utilizes complete gene-gene relationships. Unlike existing work, scCR exploits both associating and dissociating relationships, which contain valuable biological information. Given highly noisy scRNA-seq data, especially having a high number of false zeros, the goal of scCR is to recover scRNA-seq data by imputing zero values. As shown in Figure 2, our proposed framework consists of three stages: pre-imputation, complete relation, and denoising. Throughout these three stages, we enhance a gene expression matrix by gradually integrating complete gene-gene and cell-cell relationships.

4.2 Pre-Imputation Stage

We consider a cell-gene matrix X ∈ R C × G , where C and G represent the number of cells and genes, respectively. We let A cell ∈ { 0 , 1 } C × C denote an adjacency matrix of a cell-cell graph. Similarly, we let A gene ∈ { 0 , 1 } G × G denote an adjacency matrix of a gene-gene graph. Building a k -NN graph directly on X can lead to performance degradation in downstream tasks due to the noisy nature of X . Therefore, scCR begins with the pre-imputation stage, which creates a pre-imputed matrix to be used for k -NN graph construction in the complete relationship stage. In this stage, scCR imputes zero values through intercellular ( i.e. , cell-cell) propagation.

̸

Cell-cell FP. We first construct a cell-cell k -NN graph by A cell (1) = k NN ( X ) . scCR then employs FP to impute zero values by the diffusion of nonzero values among cells. We let M X ∈ { 0 , 1 } C × G be a mask matrix with M X i,j = 1 iff X i,j = 1 and M X i,j = 0 otherwise, which indicates the positions of the nonzero features in X to be preserved during the diffusion. Cell-cell FP using A cell (1) is performed as follows,

where X (1) ∈ R C × G is an output of the pre-imputation stage, which is utilized in the following complete relationship stage.

4.3 Complete Relationship Stage

In the complete relationship stage, we refine X (1) through gene-gene and cell-cell propagation. ScBFP [12] adopts gene-gene FP on the k -NN graph constructed based on cosine similarity. However, it overlooks two key issues: (1) The similarity-based gene-gene k -NN graph can connect only associating (or co-occurring) genes, excluding highly correlated dissociating (or avoiding) genes, which can offer important biological information for imputation. This occurs because associating genes may have high cosine similarity due to their co-occurrence. These genes with high cosine similarity may become connected in the cosinesimilarity-based k -NN graph . (2) As shown in Figure 3, since each gene has the distinct distribution within a gene expression matrix [15], the value distribution for a gene varies significantly among genes. Although imputation methods for scRNA-seq generally normalize a gene expression matrix in a cell-wise manner, varying scales across genes still remain after the cell-wise normalization. Therefore, propagation across genes may degrade accurate recovery by mixing values with different scales.

High cosine similarity

Strongly dissociating genes

→ connected in a k-NN graph

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Concatenation. To address these key issues, we propose a novel propagation scheme called gene-gene standardized FP. The gene-gene standardized FP first produces X com ∈ R C × 2 G by concatenating X (1) and its negative matrix along columns, i.e. , X com = [ X (1) , -X (1) ] .

Standardization. Subsequently, to enable every gene to have the same scale during propagation among genes, we standardize X com in a column-wise manner. For b ∈ { 1 , . . . , 2 G } , we standardize X com to ˜ X com ∈ R C × 2 G as follows,

Here, µ b and σ b denote the mean and standard deviation of the b -th column ( i.e. , gene) of X com . Since all the columns in ˜ X com are standardized, SFP can effectively perform propagation-based imputation without the mixing of values at various scales, addressing the aforementioned issue (2) . Furthermore, by concatenating X (1) with its negative matrix before the construction of ˜ X com , SFP can connect not only associating but also dissociating gene-gene relationships. As demonstrated in Figure 4, assume that the i -th gene and the j -th gene have strong dissociating relationships, where i, j ∈ { 1 , . . . , G } . Within any a -th cell ( a ∈ { 1 , . . . , C } ), X (1) a,i will be very high when X (1) a,j is very low and vice versa. After the standardization, the cosine similarity between the i -th gene and the j -th gene will have a large negative value, which cannot be connected in a k -NN graph. However, through the concatenation, ˜ X com : , ( G + j ) corresponds to -X (1) : ,j . Thus, the cosine similarity between ˜ X com : ,i and ˜ X com : , ( G + j ) has a large positive value, which will be connected in a k -NN graph. Therefore, through gene-gene propagation using this k -NN graph constructed by using ˜ X com , we can effectively exploit complete gene-gene relationships, including associating and dissociating relationships.

Gene-gene FP. We build a gene-gene k -NN graph on ( ˜ X com ) ⊤ by A gene = k NN (( ˜ X com ) ⊤ ) where A gene ∈ R 2 G × 2 G . We then perform the gene-gene standardized FP using A gene as follows:

where X com ∈ R C × 2 G . Since each gene in X com does not have its original scale due to the standardization, we return all the columns in X com to their original scale as follows:

where ˇ X com ∈ R C × 2 G is the rescaled matrix. We then reduce ˇ X com by retaining the first G columns, and we denote this reduced matrix as X ∗ ∈ R C × G . X ∗ is a final output of the gene-gene standardized FP.

Cell-cell FP. X ∗ containing information from complete gene-gene relationships plays a crucial role in scCR by contributing the formation of all subsequent cell-gene matrices. To perform additional

cell-cell FP using complete gene-gene relationships inherent in X ∗ , we construct cell-cell a k -NN graph by A cell (2) = k NN ( X ∗ ) . We perform cell-cell FP using X and A cell (2) as follows,

where X (2) is an output of this cell-cell FP.

Weighted sum. X (3) , which is a final output of complete relationship stage is produced by the weighted sum of X (2) and X (1) as follows:

where 0 < α < 1 is a hyperparameter. X (3) can incorporate valuable biological information since complete gene-gene relationships are delivered by X (2) .

4.4 Denoising Stage

Cell-cell Soft FP. While the pre-imputation and complete relationship stages focus on imputing zero values, the denoising stage aims to remove noise in the overall values of X via propagation-based smoothing. To exploit complete gene-gene relationships, the denoising stage utilizes X (3) containing them. We first build a cell-cell k -NN graph by A cell (3) = k NN ( X (3) ) . To denoise X , we adopt Soft FP [11] that does not maintain zero values during propagation. We apply Soft FP [11] to X as follows,

where K is the total number of propagation steps, X (4) (0) = X , X (4) ( t ) ∈ R C × G is the updated cell-gene matrix after t propagation steps, and 0 < β < 1 is a hyperparameter. An output of the denoising stage, denoted by X (4) ( K ) , is obtained after K steps.

Weighted sum. The final output of our scCR, ̂ X , is the weighted sum of X (3) and X (4) ( K ) as follows:

where and 0 < γ ≤ 1 is a hyperparameter. In summary, unlike existing propagation-based methods, our scCR enables the use of complete gene-gene relationships in denoising scRNA-seq.

BaronMouse

Pancreas

MouseBladder

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Worm Neuron

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o

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5 Experiments

5.1 Experimental Setup

We performed comparative evaluation of scCR on six widely used scRNA-seq datasets with goldstandard cell type information: Baron Mouse [25], Pancreas [26], Mouse Bladder [27], Zeisel [2], Worm Neuron [28], and Baron Human [25]. We compared our scCR with eight state-of-the-art methods handling noise in scRNA-seq data: (1) non-graph-based methods: DCA [17] and AutoClass [18]; (2) GNN-based methods: scTAG [21], scGNN 2.0 [19], and scGCL [20]; (3) propagation-based methods: MAGIC [22], scFP [11], and scBFP [12]. We evaluated scTAG only on cell clustering, since scTAG is a clustering method. To evaluate the cell clustering of scCR and baselines, we utilized three standard evaluation metrics: Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and Clustering Accuracy (CA). We used KMeans clustering to compare the clustering performance of various imputation methods, including our scCR. For dropout recovery, we employed two standard evaluation metrics: Root Mean Square Error (RMSE) and Median L1 Distance.

Implementation. For fair comparisons, we set the hyperparameters of baselines according to specifications in the papers and official codes. We reported the average performance across three independent runs. Experimental details regarding datasets, baselines, evaluation metrics, and hyperparameter settings are provided in Appendix G.

5.2 Results

scCR enables improved cell clustering. To validate the effectiveness of scCR in cell clustering, we evaluated its clustering performance. Table 1 presents the performance comparison. While FP-based baselines, including scFP and scBFP, outperformed other baselines, scCR delivered the best or competitive cell clustering performance across all datasets. Specifically, scCR improved ARI by 25 . 3% , 1 . 7% , 6 . 4% , and 21 . 6% compared to previous state-of-the-art results on Baron Mouse, Mouse Bladder, Zeisel, and Baron Human, respectively.

scCR effectively recovers dropout values. Since dropouts can occur at various rates, we generated false zeros ( i.e. , dropouts) at non-zero values in datasets by applying varying dropout rates. As shown in Figure 5, scCR significantly improved dropout recovery performance in various dropout rates across the datasets. We confirmed that scCR effectively reduced RMSE between imputed values and their original values with large reduction rates, only except for the Pancreas dataset with 40% dropout. We provided a recovery performance comparison, measured by the median L1 distance in Appendix H.2, which also demonstrates the effectiveness of scCR.

Raw

DCA

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CellType

Acinar

·Activatedstellate

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UMAP2

UMAP2

UMAP2

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: Associating

: Dissociating

scCR identifies rare cell types well. To verify the effectiveness of scCR in identifying rare cell types, we conducted in-depth analysis using the Baron Human dataset. We visualized the two-dimensional UMAP [29] repesentations of the raw data and the data with an 80% dropout rate applied. As shown in the first row in Figure 6, scCR effectively identified rare cell types with few cells, such as 'gamma' and 'Macrophage'. It is noteworthy that even under severe dropouts, scCR separated cell types well (in the second row), whereas compared methods failed.

Does scCR really model dissociating relationships? To show that scCR models both associating and dissociating gene-gene relationships, we first investigated the ratio of dissociating relationships to associating relationships. For this investigation, we define that associating or dissociating relationships exist when the absolute values of the correlation coefficients between genes exceed the top 25% . We determine that a positive sign in the correlation coefficients indicates associating relationships, while a negative sign indicates dissociating relationships. We then measured the percentages of

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associating and dissociating relationships within gene-gene k -NN graphs in scBFP and our scCR. As shown in Figure 7, scBFP hardly models dissociating gene-gene relationships. In contrast, scCR effectively models dissociating relationships, enabling the use of complete gene-gene relationships in its imputation.

scCR is even faster than existing imputation methods. To show the advantage of scCR in terms of imputation time, we measured the running time of scCR and imputation methods on datasets. As shown in Figure 8, scCR showed the lowest running time across all the datasets, regardless of the number of cells.

An ablation study (See Appendix C), further experimental results (See Appendix H), and the proof of convergence of FP (See Appendix B) are provided in Appendix.

6 Conclusion

In this paper, we proposed a novel imputation framework called Single-Cell Complete Relationship (scCR) for scRNA-seq data imputation. scCR utilized complete gene-gene relationships by concatenating a given cell-gene matrix with its negation and facilitated effective gene-gene propagation through the standardization of the cell-gene matrix in a gene-wise manner. These processes, grounded in genetic evidence, led to significant performance improvements over state-of-the-art methods in various downstream tasks on scRNA-seq data, with fast imputation times. Furthermore, our work is not limited to simply utilizing genetic evidence to design a framework. We validated this evidence within real scRNA-seq datasets, and confirmed that our scCR effectively leveraged this insight. Like other scRNA-seq imputation methods, scCR is specifically designed for scRNA-seq data. However, since scRNA-seq data are inherently matrix-formatted, scCR can be extended to general tabular data imputation. We expect that the concatenation process will be effective even for general tabular data, as there are often both positive and negative correlation coefficients between channels in such data. The extension of scCR to other domains is left for future work.

Broader Impacts

Our work provides an important insight that, when applying machine learning to the biomedical domain, it is crucial to approach with biological grounding rather than focusing solely on applying existing cutting-edge machine learning techniques. Since scRNA-seq has opened a new frontier for understanding biological systems [15, 30, 31], we believe that our work will contribute to the biomedical domain by enhancing the analysis of human diseases and the discovery of new genetic observations. We have not identified any negative impacts of our work on society.

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A Feature Propagation

Assume that a given graph G = ( V , E ) has a feature matrix X ∈ R N × F with missing values, where rows and columns correspond to nodes and F feature channels, respectively. We use A ∈ R N × N to denote a normalized adjacency, which is an input for feature propagation (FP). To preserve known features during the diffusion process, we mark the positions of the features to be preserved with 1 in the mask matrix M ∈ { 0 , 1 } N × F . Here, values of 1 in M indicate the location of observed features, where feature values will be preserved.

To formally explain the diffusion process of FP in detail, we temporarily reorder nodes for notational convenience. Since observed values may differ across channels, we reorder the nodes for each channel. When we consider the a -th channel, based on the values of 1 in M : ,a , we let V a k be the set of nodes whose a -th values are known (observed). Similarly, by examining zero values in M : ,a , V a u denotes the set of nodes whose a -th values are unknown (missing). By reordering the nodes in the order of V a k and V a u , the entire feature values and the adjacency matrix for the a -th channel can be expressed as

where x a is a column vector representing features for the a -th channel in X in the order of V a k and V a u . Here, x a k ∈ R |V a k | and x a u ∈ R |V a u | . Similarly, A ( a ) consists of four sub-matrices related to V a k and V a u . It is noteworthy that although A ( a ) ∈ R N × N and A ∈ R N × N is different due to reordering, they represent the same graph structure.

To preserve observed values during the diffusion process, we replace the first |V a k | rows in A ( a ) with one-hot vectors indicating V a k . Consequently, we obtain a transition matrix ˜ A ( a ) ∈ R N × N expressed by

where I nn ∈ R |V a n |×|V a n | is an identity matrix and 0 nz ∈ R |V a n |×|V a z | is a zero matrix. The diffusion process of FP is implemented by iterative propagation steps utilizing ˜ A ( a ) as

where ¯ x a ( t ) is an imputed feature vector after t propagation steps and 0 a u denotes a zero vector of the same length as |V a k | . As K →∞ , this recursion converges and ¯ x a ( t ) reaches a steady state (the proof can be found in Appendix B). We use ¯ x a ( K ) with large enough K to approximate the steady state.

After this diffusion process for the entire channels, we attain { ¯ x a ( K ) } F a =1 . Since these vectors have different ordering due to channel-wise reordering, we rearrange { ¯ x a ( K ) } F a =1 in the original order and construct X ∈ R N × F by stacking the originally ordered vectors in { ¯ x a ( K ) } F a =1 along the channels. In summary, FP fills in missing values in X through diffusion using A while preserving features corresponding to values of 1 in M .

B Proof of Convergence of Feature Propagation

Feature propagation (FP) [23] utilize symmetrically normalized transition matrix for the diffusion process implemented by iterative propagation steps. We prove the convergence of this diffusion process as follows.

Proposition 1. The transition matrix for the a -th channel is defined by

where ˜ A ( a ) is symmetrically normalized. Using ˜ A ( a ) , the diffusion process in the a -th channel is defined by

Then, lim K →∞ ¯ x ( a ) ( K ) converges.

The proof of Proposition 1 refers to [23]. We begin with two lemmas.

Lemma 1. A ( a ) is the symmetrically normalized matrix calculated by A ( a ) = ( D ( a ) ) -1 / 2 A ( a ) ( D ( a ) ) -1 / 2 where D ( a ) is a diagonal matrix that has diagonal entities D ( a ) ii = ∑ j A ( a ) i,j . A ( a ) uu is the | ¯ x ( a ) u | × | ¯ x ( a ) u | bottom-right submatrix of A ( a ) and let ρ ( · ) denote spectral radius. Then, ρ ( A ( a ) uu ) < 1 .

Proof. Consider A ( a ) uu 0 ∈ R N × N , where the bottom right submatrix is equal to A ( a ) uu and all other elements are zero. That is,

where 0 kk ∈ { 0 } | ¯ x ( a ) k |×| ¯ x ( a ) k | , 0 ku ∈ { 0 } | ¯ x ( a ) k |×| ¯ x ( a ) u | , and 0 uk ∈ { 0 } | ¯ x ( a ) u |×| ¯ x ( a ) k | . Given that A ( a ) represents the weighted adjacency matrix of the connected graph G , A ( a ) uu 0 ≤ A ( a ) element-wise and A ( a ) uu 0 = A ( a ) . Furthermore, considering that A ( a ) uu 0 + A ( a ) constitutes the weighted adjacency matrix of a strongly connected graph, we can conclude that A ( a ) uu 0 + A ( a ) is irreducible based on Theorem 2.2.7 in [32]. Consequently, applying Corollary 2.1.5 in [32], ρ ( A ( a ) uu 0 ) < ρ ( A ( a ) ) . Furthermore, ρ ( A ( a ) ) ≤ 1 since we can write A ( a ) = I -( D ( a ) ) -1 / 2 A ( D ( a ) ) -1 / 2 , where ( D ( a ) ) -1 / 2 A ( D ( a ) ) -1 / 2 has eigenvalues in the range [0 , 2] [33]. Note that since both A ( a ) uu 0 and A ( a ) uu share the same non-zero eigenvalues, it follows that ρ ( A ( a ) uu 0 ) = ρ ( A ( a ) uu ) . Ultimately, combining these inequalities leads to the result ρ ( A ( a ) uu ) = ρ ( A ( a ) uu 0 ) < ρ ( A ( a ) ) = 1 .

̸

Lemma 2. I uu -A ( a ) uu is invertible where I uu is the | ¯ x ( a ) u | × | ¯ x ( a ) u | identity matrix.

Proof. Since 1 is not an eigenvalue of A ( a ) uu by Lemma 1, 0 is not an eigenvlaue of I uu -A ( a ) uu . Thus I uu -A ( a ) uu is invertible.

We now prove Propostion 1 as follows.

Proof. The recursive relation can be written as

Since ¯ x ( a ) k ( t ) = ¯ x ( a ) k ( t -1) in the first | ¯ x ( a ) k | rows, it follows that ¯ x ( a ) k ( K ) = . . . = ¯ x ( a ) k . That is, ¯ x ( a ) k ( K ) retains the values of x ( a ) k . Therefore, lim K →∞ ¯ x ( a ) k ( K ) converges to x ( a ) k .

We now focus solely on the convergence of lim K →∞ ¯ x ( a ) u ( K ) . When we unroll the recursion for the last | ¯ x ( a ) u | rows,

By Lemma 1, lim K →∞ ( A ( a ) uu ) K = 0 . Therefore, lim K →∞ ( A ( a ) uu ) K ¯ x ( a ) u (0) = 0 , regardless of the initial state for ¯ x ( a ) u (0) . (we replace ¯ x ( a ) u (0) with a zero column vector for simplicity.) Hence, our focus shifts to lim K →∞ ( ∑ K -1 t =0 ( A ( a ) uu ) t ) A ( a ) uk x ( a ) k .

Given that Lemma 1 establishes ρ ( A ( a ) uu ) < 1 , and Lemma 2 affirms the invertibility of ( I uu -A ( a ) uu ) -1 , the geometric series converges as follows

In conclusion, the recursion in FP converges.

C Ablation Study

We conducted an ablation study to analyze the effectiveness of each component in scCR. We performed cell clustering on the Baron Mouse, Zeisel, and Baron Human datasets. As shown in Table 2, both concatenation and standardization contributed to performance improvement, and the combination of the two components led to significant performance improvement.

We conducted an additional ablation study to assess the effectiveness of each stage of scCR. Table 3 presents the results of the ablation study in terms of cell clustering, measured by ARI. As shown in the table, adding the complete relation stage and the denoising stage significantly improved performance compared to using only the pre-imputation stage. These results confirm that the complete relation and denoising stages contributed substantially to the high performance of scCR, underscoring the well-founded design of our approach.

D Missing Not at Random Settings

Existing studies [11, 12] simulate dropout by randomly sampling non-zero values in a cell-gene matrix from a uniform distribution and setting them to zero ( i.e. , missing completely at random (MCAR)). However, in real scRNA-seq data, dropouts occur more frequently in genes with low expression levels rather than those with high variance [34]. This is because the probability of capturing RNA transcripts of low-expression-level genes during sequencing is lower. Based on this dropout pattern, we selected the 1,000 genes with the lowest expression levels and simulated dropout only in these genes. We randomly sampled non-zero values of these genes from a uniform distribution and replaced the sampled values with zero ( i.e. , missing not at random (MNAR)).

Table 4 presents the performance comparison under the aforementioned MNAR settings in terms of data recovery, measured by RMSE. We compared our scCR to the two most competitive baselines, scFP [11] and scBFP [12]. The dropout rate was set to 20% of the total number of values in the cell-gene matrix. As shown in the table, scCR outperformed the compared methods by significant margins in the realistic dropout settings, demonstrating the robustness of scCR in realistic scenarios. Considering realistic dropout simulation can help pre-assess the generalizability of techniques in practical scRNA-seq applications.

E Memory Usage Analysis

We analyzed the memory complexity of all methods used in this paper and conducted additional experiments to examine the memory usage of our scCR. Table 5 compares the input and memory complexity of scCR with other state-of-the-art methods. To alleviate the high memory demands during k -NN graph construction, we adopted the batch-wise k -NN graph construction strategy from [12]. When constructing k -NN graphs among genes, we divided the genes into batches of size B and computed k -nearest neighbors for each batch. We applied the same batch-wise strategy when constructing k -NN graphs among cells. This approach reduces memory requirements by avoiding the need to store distances between all points in the entire dataset simultaneously. Specifically, in the memory complexity of scCR, batch-wise k -NN graph construction changes O ( G 2 ) and O ( C 2 ) to O ( BG ) and O ( BC ) , respectively. Consequently, batch-wise k -NN graph construction enables

processing of large datasets that would otherwise be infeasible due to memory constraints. Moreover, scCR does not require any trainable parameters, unlike other deep-learning-based methods.

We further measured the memory usage of scCR across various datasets, as shown in Table 6. The results in the table indicate that the advantages of scCR extend beyond its superior performance and time efficiency, showcasing its scalability as well.

F Hyperparameter Sensitivity

We conducted additional experiments to provide a comprehensive analysis of the impact of different hyperparameters, including α , β , γ , and k , on the performance of scCR. We report ARI in cell clustering on three datasets by varying α , β , γ , and k within the ranges of {0.01, 0.05, 0.1, 0.5,

0.9}, {0.1, 0.5, 0.9, 0.95, 0.99, 0.999}, {0.001, 0.01, 0.05, 0.1, 0.5, 0.9}, and {1, 2, 3, 5, 10, 15}, respectively. When varying a target parameter, other hyperparameters were fixed to their default settings. Table 7, Table 8, Table 9, and Table 10 illustrate how these hyperparameter choices impact scCR performance. As shown in the tables, the values of α , β , γ , and k used in this study generally resulted in strong performance.

In terms of sensitivity, when the runner-up ARI scores are 0 . 660 ± 0 . 00 , 0 . 848 ± 0 . 00 , and 0 . 677 ± 0 . 00 for Baron Mouse, Zeisel, Baron Human, respectively, scCR demonstrated robustness against hyperparameter variations. Specifically, α ∈ { 0 . 05 , 0 . 1 , 0 . 5 } , β ∈ { 0 . 99 , 0 . 999 } , and γ ∈ { 0 . 001 , 0 . 01 } yielded state-of-the-art performance across the datasets. For k , scCR showed strong performance across all values, except in the case of Baron Human. Additionally, some parameter adjustments led to performance surpassing that of the default settings. However, given the unsupervised nature of single-cell analysis, we retain default hyperparameter settings that generally perform well.

G Experimental Details

G.1 Implementation Details

We conducted all the experiments on a single NVIDIA GeForce RTX 2080 Ti GPU and an Intel Core I5-6600 CPU at 3.30 Hz. The number of neighbors ( k ) in cell-cell and gene-gene k -NN graphs were set to 15 and 2 , respectively. The total number of propagation steps K was set to 40 for both cell-cell and gene-gene FP. We set α , β , and γ to 0 . 05 , 0 . 99 , and 0 . 01 , respectively. We found that the choice between row-stochastic normalization and symmetric normalization applied to A cell (3) within Soft FP [11] affected performance, and we reported the best result. For dropout recovery, we excluded the denoising stage ( i.e. , γ = 1 ).

G.2 Datasets

For our experiments, we utilized six real scRNA-seq datasets, including Baron Mouse [25], Pancreas [26], Mouse Bladder [27], Zeisel [2], Worm Neuron [28], and Baron Human [25]. Table 11 summarizes the dataset statistics. We downloaded all the datasets from the GitHub repository 2 for [11]. These publicly available datasets and the repository have no public declaration of license.

We leveraged a commonly used pre-processing procedure for scRNA-seq data as described in recent studies [11, 12]. We performed minimal quality control (QC) using SCANPY [35], a toolkit for scRNA-seq analyses. Cells and genes exhibiting no gene expression ( i.e. , with all zero values) were removed from a given cell-gene matrix. We retained the 2,000 genes with the highest variance in each dataset. We then normalized each cell by total counts over all genes to ensure that every cell had an equal total count of 1 . 0 . That is, every row vector was divided by its library size, which is the sum of its values. After scaling by the median library size, log( x +1) transformation was applied to all values in the cell-gene matrix, resulting in a pre-processed cell-gene matrix.

G.3 Baselines

For all the baselines, we used the code released by the author of the respective papers. Table 12 shows the URL links for the baselines. scTAG and scGNN 2.0 are under the MIT license. The licenses of DCA, AutoClass, and MAGIC are Apache-2.0, GPL-3.0, and GPL-2.0, respectively. The code for scGCL, scFP, and scBFP has no public declaration of license. For each baseline, we adhered the hyperpameter/parameter setting in the released code or its respective paper.

G.4 Evaluation Metrics

G.4.1 Clustering

Higher ARI, NMI, and CA indicate better performance in cell clustering.

ARI. The Adjusted Rand Index (ARI) is the corrected-for-chance version of the Rand Index (RI). RI is computed as follows:

where TP is the number of true positives and TN is the number of true negatives. True positive indicates the number of cell pairs correctly assigned to the same cluster, and TP indicates the number of cell pairs correctly assigned to different clusters. ARI can be calculated as follows:

While RI produces a value between 0 and 1, ARI can produce negative values if the index is less than the expected index.

NMI. Normalized Mutual Information (NMI) is a normalization of the Mutual Information (MI) score to scale the scores between 0 and 1 . NMI is computed as follows:

where S is ground-truth cell types, I ( · , · ) is the mutual information between two input distributions, and H ( · ) is the entropy function. Here, all logs are base-2. Higher NMI indicates the distribution of predicted cluster distribution is more similar to ground-truth cell type distribution.

where s i is the ground-truth cell type of the i -th cell, c i is predicted cluster assignment of the i -th cell, and m ( · ) is the matching function responsible for mapping predicted cluster assignments to ground-truth cell types.

G.4.2 Recovery

Lower Median L1 Distance and RMSE indicate better performance in data recovery. Consider two set X = { x 1 , . . . , x n } and Y = { y 1 , . . . , y n } , where X is the set of imputed values and Y is the set of their ground-truth values.

Median L1 Distance. Median L1 Distance is calculated as follows:

RMSE. Root Mean Square Error (RMSE) is computed as follows:

H Additional Experimental Results

H.1 Varying Scales across Genes

4000

3500

3000

2500

2000

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1000

500

0

PSMA5

CTRB2

IAPP

CELA3A

PRSS2

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H.2 Dropout Recovery

Pancreas

0.85

1

0.7

0.7

0.8

4

Distance

0.5

0.55

0.6

口

0.4

口

0.4

0.3

口

D

0.25

20% -5.7%)

80%

(-6.2%) 0.2

20%

合

(-0.4%)

40%

(-4.4%)

80%

0.1 (-11.7%)

20%

(-9.4%)

40%

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80%

8

Zeisel

Worm Neuron

Baron Human

Median

1.2

0.8

1

0.95

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6

0.7

口

0.6

A

0.45

口

0.4

+

(-8.9%)

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2.(-7.7%)

(-7.6%)

(-10.4%)

0.2

(-9.2%)

(-10.8%)

0.2 (-0.4%)

(-1.2%)

(-7.4%)

20%

40%

80%

20%

40%

80%

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40%

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Dropout rate

Dropout rate

Dropout rate

口

DCA

AutoClass

A

ScGNN 2.0

A

ScGCL

MAGIC

o

ScFP

ScBFP

ScCR

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We clarified the scope of our work in the abstract and Sec. 1. The contributions of our paper are summarized in Sec. 1.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

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Question: Does the paper discuss the limitations of the work performed by the authors?

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Justification: We discussed the limitations of our work in Sec. 6 as follows: Like other scRNA-seq imputation methods, scCR is specifically designed for scRNA-seq data.

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Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduced a new scRNA-seq imputation method, scCR, utilizing both associating and dissociating gene-gene relationships to improve the accuracy of scRNA-seq imputation, especially on noisy dataset. \nThe method constructed comprehensive k-NN graph for both cell-cell and gene-gene. Standardize the value distribution of each gene to enhance gene-gene relationship modeling. \nComprehensive analysis on multiple datasets demonstrates the advantage of scCR compared with state-of-the-art methods in two tasks, gene expression recovery and cell clustering, especially on rare cell types.\n\nSTRENGTHS:\nThis paper tackles an important problem, scRNA-seq data imputation, with extra information from negatively correlated gene relationships. The idea itself is straightforward yet effective. \nThe paper includes extensive and solid experiments on multiple datasets. The experiments show solid performance improvement compared to multiple baseline methods. The paper explores the variance on dataset selection and dropout rate. \nThis paper also includes a scalability study to demonstrate its advantage on runtime.\n\nWEAKNESSES:\nThough the paper explores variance on dataset selection and dropout rate, yet the variance is mainly focus on dataset. It would be helpful to include a detailed sensitivity analysis of key hyperparameters, such as k for k-NN, \\alpha, \\beta, and \\gamma. \nThe paper includes a runtime analysis, but not a memory usage analysis. The paper can be more comprehensive to include memory usage analysis to show its advantage on scalability.\n\nQUESTIONS:\nThe overall paper is quite complete and comprehensive. \nOnly a few questions, 1. the impact of hyperparameter selection, additional set of experiments would be perfect. But due to the time limit, theoretical analysis would also be very helpful to understand how sensitive the model is and make the method easier to use.\n2. The memory usage of this method on different scales gives potential users a better sense of its applicability and resource requirements.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

comments: SUMMARY:
This paper introduces a novel imputation method for scRNA-seq data that accounts for both associating and dissociating gene relationships by using a k-NN graph and negating the cell-gene matrix. The method standardizes gene value distributions and shows significant performance improvements in cell clustering and gene expression recovery across six datasets, outperforming existing methods.

STRENGTHS:
1.	This paper introduces a new imputation method named Single-Cell Complete Relationship (scCR) that addresses the limitations of current propagation-based approaches for scRNA-seq data by modeling both associating and dissociating gene-gene relationships.
2.	The extensive experiments conducted by the authors show that scCR significantly outperforms existing methods in gene expression recovery and cell clustering, highlighting its effectiveness in capturing complete gene-gene relationships and improving the quality of scRNA-seq data analysis. 
3.	The paper is well-structured and clearly written.

WEAKNESSES:
1.	The proposed method appears too simple and lacks significant innovation.
2.	The paper does not clearly articulate the motivation behind the proposed method. Specifically, it does not adequately explain what dissociating relationships among genes are or why identifying these relationships is effective for addressing the research problem. 
3.	The experiments lack biological validation, and relevant case studies are needed to support the findings.

QUESTIONS:
1.	There are now many methods using large models for cell clustering, such as SCGPT and GeneFormer. How do these methods compare in terms of effectiveness?

LIMITATIONS:
None
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an approach for single-cell RNA-seq data imputation. The data comes as a matrix capturing relationship between cells and genes. Zero values in that matrix represent unobserved gene expression that can result from technical omissions (known as dropouts) and true biological absence. The non-zero values also suffer from noise such as cell and batch effects. The goal is to impute and de-noise observed single-cell RNA-seq data.\n\nAn effective prior approach is based on kNN graphs, where one first builds an adjacency matrix between cells or genes using cosine similarity and kNN neighborhoods relative to RNA-seq data. In contrast to prior work that focuses on adjacency matrices informed by “association” links, the proposed approach aims at modelling “dissociation” links that have negative cosine similarity. The approach proceeds in multiple stages: \n- Pre-imputation stage where kNN graph is built on the input matrix and similar to Markov chains, starting from random state (i.e., feature matrix of dimension $N_{cell} \\times N_{gene}$) one “diffuses” to the stationary distribution (Appendix A for details). \n- The second stage appends negated pre-imputed matrix making the feature matrix $N_{cell} \\times 2N_{gene}$ that allows for kNN graphs accounting for both “association” and “disassociation” relationships to be reflected in the adjacency matrix. This is followed by gene-to-gene and cell-to-cell propagation (i.e., stationary distribution of the Markov chain given by these matrices) with resulting matrices convexly combined into imputed “complete relationship”. \n- Final step is de-noising where now propagation of information involves convex combination with the original feature matrix and adjacency matrix build using kNN graph on the back of “complete relationship” from step ii). The final output is a convex combination of steps ii) and iii).\n\nExperiments involve cell clustering, recovery of dropout rates, robustness relative to dropouts, identification of rare cell types, modelling disassociation rates, etc.\n\nSTRENGTHS:\nThis is an interesting problem that relates nicely to link prediction in graph neural networks. It has been clearly presented with strong motivation by noise and dropouts. The paper is also clearly written and well-organized. I would hope that there will be follow up from the community focusing on link prediction.\n\nThe approach is not straightforward and involves several “propagation” steps and, contrary to past work, incorporates the disassociation and negative correlations into kNN adjacency matrices. \n\nEmpirical performance in some cell clustering tasks is not incrementally but significantly improved. The method also appears to be more robust relative to dropouts than the alternatives.\n\nWEAKNESSES:\nAn ablation is missing relative to different steps (pre-imputation, complete relationship, de-noising stage). Hence, it is unclear if they are needed and to what extent they contribute to the performance improvement.\n\nImputation metrics might be dependent on dropout strategy and it would be good to discuss what kind of “random” strategies have been used and how likely they are to reflect the corruptions specific to single-cell RNA-seq. Is there any way to generate “challenging” splits that are better at reflecting the “generalization” and \"robustness\"?\n\nQUESTIONS:\nM is of dimension N x F (line 112)?\n\nHow do you decide that cell row is “unknown” from observed data?\n\nDropout rates? How was this done exactly?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-16"}