### The application of conventional log normalization results in excessive detection of the signals

Most popular analysis tools for scRNA-seq data, including Seurat, Scanpy, and Monocle3, employ log normalization with various scaling factors, typically greater than 1000 by default, as a preprocessing step to address bias and skewness in data^{8,20,21}. They then reduce the dimensionality of data to capture true biological signals by filtering out noise^{16,18}. During this process, setting a threshold for distinguishing signal from noise is crucial. However, in most cases, this decision is left to the user^{8,20,21,33}. To circumvent this subjective choice, we employed a recently developed noise filtering method based on the random matrix theory (RMT-based noise filtering)^{32}. RMT-based noise filtering provides a data-driven threshold that distinguishes biological signals from random noise in the data (Fig. 2a). Specifically, we first preprocessed the data using log normalization. Then, by multiplying the log-normalized data by its transpose, we calculated the cell similarity matrix (Fig. 2b). Subsequently, using the Eigenvalue Decomposition (EVD) algorithm, a comprehensive set of eigenvalues of the cell similarity matrix was obtained (Fig. 2c). These values were fitted to a Marchenko-Pastur (MP) distribution, a universal distribution of eigenvalues obtained from random matrices. Eigenvalues conforming to the MP distribution are considered to be noise from a random matrix, while those deviating from the MP distribution, surpassing the threshold of the Tracy-Widom (TW) distribution (vertical line in Fig. 2c), are considered to be potential biological signals^{32,41}. Using 33 detected signal eigenvalues and their corresponding eigenvectors, i.e., signal vectors (matrix shown in Fig. 2c), low-dimensional data can be obtained. However, the 2D embedding obtained with UMAP on the low-dimensional data failed to capture the high-dimensional data structure precisely (Fig. 2d). We suspected this inaccuracy arises from the excessive number of detected signals, which is overly abundant for differentiating merely three clusters (Fig. 2c).

### L2 normalization prevents signal distortion due to conventional log normalization

To find out the cause of this excessive signal detection, we constructed a 2000 by 2000 pure noise random matrix with elements drawn from a Poisson distribution with a mean of 2 (Fig. 2e). When log normalization was applied to the matrix, all eigenvalues of its cell similarity matrix (Fig. 2f) followed the MP distribution, i.e., no signal was detected (Fig. 2g). This can be understood from the cell similarity matrix (Fig. 2f), whose elements are defined as the inner product between two cell vectors, with each vector being an array of normalized gene expression levels from a single cell. As a result, the diagonal elements in this matrix represent the square of the lengths of the cell vectors, which are comparable to each other due to log normalization (Fig. 2f). This general structure allows off-diagonal elements in the matrix to be interpreted as the directional similarity between cell vectors. Thus, off-diagonal elements close to zero in the matrix (Fig. 2f) indicate no substantial directional similarity between cells, explaining the absence of signal (Fig. 2g). Next, we concatenated the dense random matrix (Fig. 2e) and the sparse binary matrix with the size 2000 by 8000 to reflect the high sparsity of scRNA-seq (Fig. 2h). When log normalization was applied to this sparse random matrix, no signal was detected again, indicating that signal distortion does not occur even when data shows high sparsity (Fig. 2i, j).

To further reflect the bias in the cell’s TGC in scRNA-seq data, we divided the first 400 rows of the dense part of the random sparse matrix by two, resulting in a cell group of low TGC (Fig. 2k). Such bias in the TGC, i.e., the differences in sequencing depth, was expected to be removed by the log normalization. However, unexpectedly, ~400 signals, surpassing the TW threshold, were detected (Fig. 2m). As the number of detected signals matches the number of cells with low TGC, we hypothesized that cells with low TGC are associated with artificial signals. We noticed that 400 diagonal elements of the cell similarity matrix corresponding to cells with low TGC were much larger than the other diagonal elements (Fig. 2l, red box), unlike the cell similarity matrix of the dense random matrix (Fig. 2f) and the sparse random matrix (Fig. 2i). This means that the lengths of 400 cell vectors with low TGC are much longer than that of the other 1600 cell vectors. As a result, the inner products with the long 400 cell vectors (Fig. 2l, red box) became larger than those with the other cell vectors (Fig. 2l, outside of the red box). In short, the cell similarity matrix no longer accurately reflects the directional similarity among the cell vectors. In particular, the increased inner products with the low TGC vectors, caused by the exaggerated lengths of the low TGC vectors (Fig. 2l), created an artificial directionality toward the low TGC vectors. This explains why the 400 eigenvalues of the cell similarity matrix surpassed the TW threshold (Fig. 2m). Such distortion of the signal became higher as higher levels of sparsity and TGC variance were introduced to random matrix (Supplementary Fig. 1).

The signal distortion occurs because log normalization fails to introduce uniformity to cell vector lengths when a matrix is highly sparse and has a bias in TGC, which is typical of scRNA-seq data^{14,16,18,19}. This is surprising because during the first step of log normalization, library size normalization eliminates cell-specific bias in data by dividing gene expression in each cell by its TGC (i.e., step normalizes the cell vector lengths)^{15,16,18,21}. However, we found that this library size normalization is disrupted by the follow-up gene scaling step because it over-amplifies non-zero values in genes containing many zeros, especially in cells with low TGCs (Supplementary Fig. 2).

Thus, to uniformize the lengths of cell vectors, we added L2 normalization after gene scaling. Although L2 normalization is a very simple approach, it removed all artificial signals effectively (Fig. 2n, o, Supplementary Fig. 1). Next, we tested whether L2 normalization can improve the low-dimensional embedding in Fig. 2d. When L2 normalization was applied to the log-normalized data, cell vector lengths became identical (Fig. 2p), and thus the number of detected signals was considerably reduced, from 33 (Fig. 2c) to 6 (Fig. 2q). This yielded 2D embedding (Fig. 2r), which more accurately portrays the high-dimensional data structure compared to previous one (Fig. 2d). However, overlaps between clusters still existed in 2D embedding, limiting resolution (Fig. 2r).

### Signal robustness test filters out low-quality signals due to non-biological zeros

Despite considerable improvement after L2 normalization, we still observed some overlap between the sub-clusters (Fig. 3a–c). We hypothesized that this sub-optimal result stems from the noise associated with the biologically irrelevant zeros because some zeros in scRNA-seq data are caused by stochastic dropout events rather than biological zeros. To handle biologically irrelevant zeros, various imputation methods have been developed^{25,26,27,28,29}. However, every imputation technique unavoidably alters the original data, potentially leading to the misinterpretation of the biological information contained within the data^{17,28,30,31}. Thus, we developed an alternative method that preserves the original data while filtering out low-quality signals contaminated by biologically irrelevant zeros.

We found that the low-quality signals mainly stem from low expression of genes that do not share a strong common expression pattern across cells (Supplementary Fig. 3). Thus, they are expected to be susceptible to slight perturbation of data, which can mask the spurious correlations of sparse genes. To introduce slight perturbations to the original data, we generated a binary (0 or 1) random matrix with a sparsity level of 0.97 or greater and added it to the original count matrix (Fig. 3d). We then quantified how much the signal vectors (Fig. 3e) are perturbed by calculating the absolute inner product between all signal vectors from the unperturbed data and all eigenvectors from the perturbed data (Fig. 3f (i)). Next, we constructed the column-wise maximum vector of the inner product matrix (Fig. 3f (ii)), whose (i)-th component represents the inner product between the (i)-th signal vector and its most similar eigenvector from perturbed data. We obtained multiple column-wise maximum vectors by repeating the process with different perturbation matrices and then used their mean vector to indicate the stability of signal vectors (Fig. 3f (iii)). In particular, a large (i)-th component of the mean vector indicates that the (i)-th signal vector is robust against the data perturbation (Fig. 3f (iii), red arrows). Utilizing only these three robust signals (Fig. 3f (iii), red box) among six signals detected using RMT-based signal filtering (Fig. 3b) enabled accurate distinction of three clusters without any overlaps between them in the 2D embedding (Fig. 3g). This result underscores that selecting the correct number of signals is essential for the successful downstream analysis.

### scLENS (single-cell Low-dimensional Embedding using effective Noise Subtraction)

By integrating log normalization and L2 normalization into scRNA-seq data preprocessing, along with implementing signal detection using RMT-based noise filtering and a signal robustness test, we developed a dimensionality reduction tool, scLENS (single-cell Low-dimensional Embedding using effective Noise Subtraction). To facilitate the use of scLENS, we provide a user-friendly computational package that automates dimensionality reduction, thus bypassing the need for labor-intensive and time-consuming parameter tuning (see Supplementary Information for manuals).

We evaluated the performance of scLENS on real data using ZhengMix data^{42}, which consists of purified peripheral blood mononuclear cells (Fig. 3h (i)). This dataset has been utilized for various benchmarking studies since the data include true labels, but it is challenging to classify the cell types due to high sparsity and variation in the TGC^{43,44}. Upon applying conventional log normalization to the data, T-cell subtypes were not clearly distinguished in the 2D embedding constructed from 84 detected signals detected by RMT-based noise filtering (Fig. 3h (ii) dashed circle). RMT-based noise filtering detected a reduced number of 42 signals after applying L2 normalization following log normalization, leading to the further refinement of the 2D embedding (Fig. 3h (iii) dashed circle). This embedding was more improved by selecting 13 robust signals from 42 signals detected by RMT-based noise filtering, using signal robustness test-based filtering (Fig. 3h (iv) dashed circles). While this result is automatically obtained via scLENS without any parameter tuning, the result is comparable with the best result of ZhengMix data obtained from massive parameter tuning with various DR methods (Supplementary Fig. 4)^{44}.

### scLENS excels at handling sparse data with high TGC variance

Next, we benchmarked scLENS with the other 11 popular packages with their default settings (see Supplementary Table 1 for details). Among these, well-known packages like Seurat, Scanpy, and Monocle3 employ log normalization for preprocessing and Principal Component Analysis (PCA) with 50 principal components (PCs) by default for DR. Unlike these methods, ParallelPCA (Horn’s parallel analysis) of PCAtools automatically selects PCs based on their statistical significance against those of randomized data^{45,46}. Similar to ParallelPCA, Randomly^{32} also automatically selects the signals based on RMT and employs log normalization as preprocessing. On the other hand, DR methods implementing matrix factorization (ACTIONet)^{33}, random projection (SHARP)^{38}, and autoencoder (scDHA and scVI)^{27,39} rather than PCA generally exclude the gene scaling step in log normalization during the preprocessing. Furthermore, scDHA^{37} and SHARP^{38} do not even use the library size normalization step, which is the first step of log normalization. We also examined ZINB-WaVE^{40}, employing a model-based DR method, and scTransform^{23}, using a model-based normalization method, as alternatives to log normalization.

To benchmark scLENS against other DR methods, we utilized simulation data generated by scDesign2^{47}, which produces simulated data with true labels. After training on immune cell data^{6}, we simulated approximately 60,000 simulation cells of ten T-cell subtypes. By subsampling ~3000 cells each from 60,000 simulated cells, we generated 13 datasets with different sparsity levels and the coefficient of variation (CV) of TGC.

From the 13 test datasets, we selected five datasets with similar CV values of TGC, but varying sparsity levels, to evaluate the impact of sparsity on the package’s performance. For the data with the lowest sparsity, scLENS achieved the highest silhouette score (SIL score) (Fig. 4a, left). After scLENS, three packages—Scanpy, Seurat, and Monocle3—demonstrated overall good performances, all employing log normalization (Fig. 4a, left). Unlike these three packages that use a default of 50 PCs, ParallelPCA, which automatically selects PCs, showed slightly lower performance than them, but overall, it also showed a good performance. Next, scTransform, which utilizes model-based normalization, along with ACTIONet and scVI, both using log normalization without gene scaling, exhibited intermediate performance (Fig. 4a, left). Following these packages, Randomly, which employs log normalization as preprocessing and automatically selects PCs, showed below-average performance. Two packages, scDHA and SHARP, which use only log transformation during preprocessing and model-based DR (ZINB-WaVE), showed lower performance levels (Fig. 4a, left). As sparsity increases, the SIL score of all packages decreases (Fig. 4a, left). Nevertheless, the decrease in scLENS’s performance was less than that of the other 11 packages (Fig. 4a, left).

The SIL score is a distance-based metric, so it can be sensitive to noise and outliers in the data. Thus, we used an alternative metric, the element-centric similarity (ECS), which measures the similarity between the clustering result obtained using hierarchical clustering and the ground truth label to evaluate a given DR method’s performance (see “Method” section for details). In terms of ECS, scLENS achieved the highest performance at the lowest sparsity level, which is consistent with previous results evaluated by the SIL score (Fig. 4a, right). In addition, as sparsity levels increased, scLENS demonstrated minimal degradation in ECS, whereas the other DR methods experienced significant declines in ECS (Fig. 4a, right). The more substantial performance degradation observed in DR methods using a fixed number of PCs compared to scLENS is attributable to their inability to account for the reduced biological information as sparsity increases. Furthermore, compared to these packages using a fixed number of PCs, Randomly and ParallelPCA, designed to select the PCs automatically, showed lower performance, suggesting their sub-optimal effectiveness in identifying signals in data (Fig. 4a). On the other hand, scLENS’s capability to identify the optimal number of signals, which decreases from 22 to 18 with increased sparsity level, resulted in its 2D embedding providing a more distinct separation of true cell types compared to the 2D embeddings from the three packages using a fixed 50 PCs by default (Fig. 4b). As a result, scLENS achieved the highest performance, in terms of ECS and SIL (Fig. 4a, b).

Next, to investigate the influence of variation in CV of TGC on the performance of DR methods, we selected five datasets with similar sparsity levels, but varying CVs in TGC. When CV of TGC was low, scLENS and Monocle3 achieved the highest performance, while the seven DR methods using log normalization, log normalization without gene scaling, and model-based scaling (scTransform) as preprocessing methods, demonstrated good overall performance (Fig. 4c). On the other hand, scDHA, ZINB-WaVE, and SHARP continued to display lower performance levels at the lowest CV of TGC (Fig. 4c). As CV of TGC increased, scLENS and scTransform showed no significant performance changes (Fig. 4c, right). In contrast, the performance of the seven DR methods, which use log normalization as preprocessing and a fixed number of signals by default, showed substantial decreases with an increasing CV of TGC (Fig. 4c right). Notably, when the CV of TGC was low, scLENS detected 49 signals, which is close to the default value of 50 PCs used by Scanpy, Seurat, and Monocle3. In this case, with 50 PCs by default, these three packages performed comparable to the scLENS by providing 2D embeddings showing the clear separation between true clusters. However, as the CV of TGC increased, the inter-cluster distances in the embeddings generated by Scanpy, Seurat, and Monocle3 became distorted, leading to significant overlaps between clusters (Fig. 4d). Conversely, scLENS maintained the clear separation between true clusters in its 2D embedding by effectively reducing the number of detected signals from 49 to 19 as a CV of TGC increased (Fig. 4c, d). These results emphasize the importance of accurate signal selection, especially in data with high level of sparsity and CV of TGC.

### scLENS outperforms other DR methods on data with abundant non-binary information

We evaluated scLENS across a more diverse range of data types by combining 16 real and ten simulation datasets with the previous 13 simulated T-cell datasets (Supplementary Table 2). Specifically, to broaden our benchmarking on diverse types of data that encompass various cell types, we generated 10 additional simulation datasets from Tabula Muris data^{48}, obtained from various mouse tissues, with scDesign2^{47}. Furthermore, to evaluate the performance of various benchmarking packages on real data, we included three real datasets, Koh^{49}, Kumar^{50}, and Trapnell data^{51}, whose cell labels were determined independently of the scRNA-seq assay to minimize evaluation bias towards particular analysis tools used in each study^{43}. In addition, we used thirteen real datasets generated by mixing the Zheng data^{42}, which contains eight pre-sorted blood cell types, while adjusting the number of cells and the subpopulation ratios.

For the extended datasets, scLENS outperformed all other DR methods in terms of ECS scores based on both hierarchical clustering and graph-based clustering as well as SIL score (Fig. 5a). Next, we compared the performance of scLENS and the other DR methods depending on the sparsity level, and CV of TGC of the data, in terms of ECS score obtained by applying hierarchical clustering (Fig. 5b). For each dataset, we calculated the difference between the ECS of scLENS and the highest ECS recorded by the other 11 DR methods, referred to as the relative performance of scLENS. As the sparsity of data increases, the relative performance of scLENS showed overall increases (Fig. 5b), consistent with our previous results based on simulated data (Fig. 4a). On the other hand, there is no correlation between CV of TGC and the relative performance (Fig. 5b), in contrast to our analysis based on simulated data (Fig. 4c).

Next, we investigated why there is no correlation between the CV of TGC and relative performance. Generally, the data contains two types of clustering information: binary and non-binary information. The binary information comes from the indices (positions) of the zero-valued elements within the data matrix, while non-binary information stems from the different magnitudes in non-zero values. The non-binary information is expected to be more distorted by conventional log normalization compared to the binary information because the conventional log normalization can overly amplify the gap between zero and non-zero values and reduce the variance in the non-zero values^{22} (Supplementary Fig. 6). Thus, we hypothesized that as the proportion of non-binary information decreases, the distortion of the log transformation becomes weaker and thus the relative performance of scLENS decreases. To test this hypothesis, we quantified the proportions of non-binary and binary information in the dataset. Specifically, we randomly shuffled the non-zero values to disrupt the non-binary information and then calculated the degree of change in SIL scores of 2D embeddings obtained by scLENS (ΔSIL = SIL—SIL_{p}) (Fig. 5c, Supplementary Fig. 5). As the proportion of non-binary information decreases, the effect of shuffling in non-zero values decreases, and thus ΔSIL decreases (Fig. 5c).

Indeed, the relative performance of scLENS depends on ΔSIL, measuring non-binary information (Fig. 5d). Specifically, when data contains mostly binary information and little non-binary information (i.e., low ΔSIL), scLENS’s relative performance is low regardless of CV of TGC (Fig. 5d). For example, in the simulated Tabula Muris data (Sim. T-muris No. 3 in Fig. 5d), the difference between 2D embeddings of scLENS before and after shuffling was barely noticeable (Fig. 5e). This indicates that the clustering for this data is mainly based on binary information. In such cases, scLENS and Monocle3 have no performance differences. On the other hand, for data with high non-binary information (i.e., high ΔSIL) (Fig. 5d), scLENS outperforms the other methods. For instance, in simulated T-cell data (Sim. T-cell No. 9 in Fig. 5d), the 2D embedding of scLENS is completely disrupted by the shuffling of non-zero values (Fig. 5f top), indicating that the patterns in non-zero values are critical for the clustering. In contrast, shuffling shows a little disruption of 2D embedding by Monocle3 (Fig. 5f bottom). This indicates that the embedding of Monocle3 is mainly based on binary rather than non-binary information. This occurs because conventional log normalization exaggerates binary information, thereby causing an artificial reduction in the relative portion of non-binary information in the sparse data with a high CV of TGC. This explains a recent puzzling study reporting that dimensionality reduction involving log normalization on count data generates similar low-dimensional embeddings to those obtained from binarized data^{52}. Taken together, when there is enough binary information in a dataset for clustering (easy case), scLENS and the DR methods based on conventional log normalization have similar performance (Fig. 5d, e). On the other hand, when non-binary information is critical for clustering (difficult case), scLENS outperforms the other tested DR methods (Fig. 5d, f).

### scLENS outperforms other DR methods in capturing local structure in data

So far, the performance evaluation has focused on the clustering and UMAP embedding performances of 12 packages using a limited number of real and simulated datasets with ground truths. To extend our analysis and diversify data types, we performed a downsampling benchmark approach using kNN-overlap scores, inspired by the study of Ahlmann-Eltze et al.^{15}. For this analysis, we newly collected 15 deeply sequenced UMI count datasets and four read count datasets, each characterized by an average TGC exceeding 25,000 per cell^{53,54,55,56,57,58,59,60,61,62} (Supplementary Table 2). These datasets were then downsampled to an average TGC of 5000 per cell, aligning with the typical sequencing depth of 10x genomics data. Subsequently, downsampled and original data were reduced in their dimensionality after applying 12 DR methods. We then evaluated the similarity between two kNN-graphs constructed from dimensionally reduced original and downsampled data using the average KNN-overlap score, which estimated each package’s performance in capturing the original local complex structure from downsampled data (see “Methods” section for details).

scLENS outperforms the other 11 DR methods (Fig. 6a), similar to the result of the clustering performance benchmark (Fig. 5a). Along with scLENS, scTransform and ACTIONet, which used model-based normalization and log normalization without gene scaling for preprocessing, respectively, demonstrated overall good performances (Fig. 6a). Next, Monocle3, employing log normalization for preprocessing, showed competitive performance (Fig. 6a). Compared to the Monocle3, using a fixed number of 50 PCs, two packages, ParallelPCA and Randomly, which automatically detect signaling PCs, exhibited lower performance, indicating their ineffectiveness in identifying the optimal number of signals in data (Fig. 6a). Following them, the widely used three packages, Seurat, Scanpy, and scVI, which utilize log normalization and feature selection to select highly variable genes during preprocessing, showed sub-optimal performance (Fig. 6a). Consistent with the findings in the clustering benchmark, three packages, scDHA, SHARP, and ZINB-WaVE, demonstrated lower performances in terms of the average kNN-overlap as well (Fig. 6a).

Additionally, using the average kNN-overlap score, we compared scLENS’s effectiveness in determining the optimal number of signals against two well-known methods: the elbow method and the 95% variance criterion. For this comparative analysis, we downsampled 19 original datasets from a previous downsampling benchmark with ratios ranging from 0.1 to 0.5 and normalized them using log normalization with L2 normalization. We then reduced the normalized datasets to the top 100 PCs and generated 98 kNN graphs by varying the number of PCs from 3 to 100. By comparing these graphs to the reference kNN graph, constructed by applying scLENS to the original dataset, we computed the average kNN-overlap scores for each number of PCs. From these scores, three kNN-overlap scores, corresponding to three PC sets selected by scLENS (blue star in Fig. 6b), the elbow method (red star in Fig. 6b), and the 95% variance criterion (green star in Fig. 6b), were collected. Subsequently, we normalized these scores by dividing them by the maximum average kNN-overlap score (orange star in Fig. 6b) for each dataset to facilitate comparative analysis across different downsampling ratios (Fig. 6c). In this result, we found that, as the downsampling rate increases, the 95% variance criterion’s performance increases, while the elbow method’s performance decreases significantly (Fig. 6c). This implies that the 95% variance explained criterion often identifies more PCs as signals than the optimal number, while the elbow method typically selects fewer PCs as signals in comparison to the optimal number. In contrast, scLENS consistently chooses a number of signals close to the optimal number of signals, achieving performance that closely approaches the highest peak performance (Fig. 6c).

### Median scaling enhances the speed and memory efficiency of scLENS

Although scLENS demonstrates superior performance compared to other packages, it requires substantial memory due to its use of whole cells and genes that remain after quality control (QC). Post-QC, ~10,000 genes typically remain in most datasets, with some cases exceeding 18,000 genes (Supplementary Data 1). Furthermore, from these large-sized datasets, scLENS computes the complete sets of eigenvalues and eigenvectors to fit the MP distribution, which significantly increases scLENS’s memory requirements. To reduce these memory requirements, we allocated only non-zero values and their indices in the data matrix to memory using SparseArrays.jl module. However, this approach loses its memory efficiency as a substantial increase in non-zero values occurs due to mean value subtraction during gene scaling. To address this, we modified the gene scaling step by replacing the subtraction value from the mean of each gene with their median (see Methods for details). Using the median scaling, we maintained a high level of sparsity even after the log normalization.

Indeed, scLENS with the median scaling (scLENS-med) requires lower memory than the original scLENS (Fig. 7a). Specifically, across the 39 datasets used in the clustering benchmark study, the CPU memory requirement was reduced by ~2.4 times, and GPU memory usage decreased by ~1.6 times on average (Fig. 7a). Despite the lower memory requirement, scLENS-med showed similar and slightly lower performance in terms of ECS and the average kNN-overlap score, respectively, compared to the original scLENS (Fig. 7b) across all 58 datasets used in previous benchmarking (Figs. 5 and 6). Moreover, scLENS-med was ~1.7 times faster than the original scLENS across the same datasets used in the memory performance analysis (Fig. 7c). This is noteworthy given that the original scLENS already had reasonable speed compared to the others due to utilizing GPU (Fig. 7c).

For four large datasets, each containing approximately 20,000 genes but varying in cell count from 25,000 to 100,000, scLENS-med requires reasonable memory and reasonable speed (Fig. 7d, e). That is, a dataset with 25,000 cells requires ~23GB of CPU memory and 20 min while processing 100,000 cells necessitates ~93 GB of RAM and 180 min (Fig. 7d green and Fig. 7e). This heavy CPU memory consumption occurs during preprocessing and generation of shuffled and perturbed data for the robustness test, and increases as the number of data matrix elements increases (Fig. 7d green). In contrast, the GPU memory consumption does not correlate with the number of cells in the data (Fig. 7d brown). This is because scLENS computes the same number of eigenvalues and eigenvectors as genes, usually lower than the number of cells in large datasets.

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