Zero-preserving imputation of scRNA-seq data using low-rank approximation
Abstract
Single cell RNA-sequencing (scRNA-seq) methods have revolutionized the study of gene expression but are plagued by dropout events, a phenomenon where genes actually expressed in a given cell are incorrectly measured as unexpressed. We present a method based on low-rank approximation which successfully replaces these dropouts (zero expression levels of unobserved expressed genes) by nonzero values, while preserving biologically non-expressed genes (true biological zeros) at zero expression levels. We validate our approach and compare it to two state-of-the-art methods. We show that it recovers true expression of marker genes while preserving biological zeros, increases separation of known cell types and improves correlation of simulated cells to their true profiles. Furthermore, our method is dramatically more scalable, allowing practitioners to quickly and easily recover expression of even the largest scRNA-seq datasets.
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In-depth summary
Claude Opus 5.5 · xhigh effort · 17 Oct 2026 · from the full text of version 1 (22 Aug 2018)
Truncated SVD plus a per-gene threshold at the most negative value fills dropouts while leaving biological zeros alone, and runs on 50,000 cells in about 40 minutes.
Popular dropout-imputation methods treat every zero as missing, so they also fill in genes a cell genuinely does not express, and they do not scale to datasets of hundreds of thousands of cells. ALRA exploits the observation that, in a low-rank approximation of a non-negative matrix, the reconstructed values at true zeros are symmetric about zero, so the most negative entry of each gene gives a data-driven threshold below which values are reset to zero.
Key results
SuggestedOn simulated cells with known ground truth, ALRA kept almost all genuinely unexpressed genes at zero, whereas MAGIC filled them all and SAVER filled more of them than ALRA did.
Simulating 5,000 cells by multinomial sampling from ImmGen bulk profiles of nine spleen immune populations (17,580 genes, 93% zeros), ALRA preserved 99% of true biological zeros while completing a large share of technical zeros. MAGIC completed all biological zeros (about 47% positive); SAVER was intermediate. Balanced accuracy was 80% (ALRA), 75% (SAVER) and 65% (MAGIC). · 2.1. ALRA preserves biological zeros in…; 5.3. Simulated scRNA-seq data
SuggestedImputed expression tracked RNA FISH measurements of gene-expression variability better than the compared method, and did so without the filtering steps that comparison needed.
On 8,498 melanoma Drop-seq cells with matched RNA FISH for 26 genes, Gini coefficients after ALRA correlated 0.86 with FISH on the 15 filtered, GAPDH-normalised marker genes, against 0.6 for SAVER and 0.3 for raw Drop-seq. Without SAVER's filtering and normalisation steps, ALRA still reached 0.82 on the full dataset. · 2.3 ALRA recovers true expression distribution; 5.5. FISH experiments
ShownCell types became easier to tell apart after imputation in a large cortex dataset, measured by a supervised classifier rather than by eye.
On 65,539 mouse visual cortex cells (19,155 genes) with 30 author-assigned types, a random forest's out-of-bag error fell from 6.3% on the observed data to 2.6% after ALRA (run with k = 48 on the 48,244 classified cells). SAVER could not be run on the full matrix, so the published SAVER comparison uses a 7,387-cell subsample. · 2.2. ALRA improves separation of cell types; 5.4. Improved separation of mouse cortical cells
ShownRandomised SVD made the method much faster than MAGIC and SAVER in single-core runs, and in simulation its accuracy improved as more cells were used, so scaling matters beyond convenience.
On single-core runs over cortex subsets up to 50,000 cells with 19,155 genes, ALRA took about 40 minutes at 50,000 cells, MAGIC over 4 hours at the same size, and SAVER over 12 hours at 5,000 cells (larger sizes not attempted). In simulation, incorrectly completed biological zeros per population fell and correctly completed technical zeros rose as the number of cells grew. · 2.4. ALRA is highly scalable; 5.6. Runtime comparison
ShownThe method is deliberately conservative: when the reconstructed values for expressed and unexpressed cells overlap, it leaves technical zeros uncompleted rather than risk filling biological ones.
Across 10 FACS-purified blood populations (83,992 cells, 12,776 genes after filtering, k = 20), ALRA restored the expected sparsity of the markers CR2, NCAM1, CD8A and CD4, cleanly for CR2 and NCAM1; for CD4 and CD8A the two value distributions overlapped, so some technical zeros were also set to zero. SAVER and MAGIC filled technical zeros but also many expected biological ones. · 2.1. ALRA preserves biological zeros in…; 5.2. Purified PBMC analysis
Methods, data and reproducibility
After library normalisation to 10,000 UMIs per cell and a log transform, a randomised rank-k SVD is computed, each gene is thresholded at the absolute value of its most negative reconstructed entry, survivors are rescaled to that gene's original non-zero mean and standard deviation, and originally non-zero entries pushed below threshold are restored. The rank k comes from singular-value spacings.
- FACS-purified blood cell populations from 10x Genomics, merged and filtered (cells with >400 genes, genes in >100 cells) · n = 94,655 cells x 32,738 genes, filtered to 83,992 x 12,776
- Simulated single-cell profiles drawn from a multinomial parameterised by ImmGen bulk RNA-seq of nine spleen immune cell types, giving known biological vs technical zeros · mouse · n = 5,000 cells, 17,580 genes, 93% zeros
- Mouse visual cortex scRNA-seq from Hrvatin et al., filtered as in Huang et al. · mouse · n = 65,539 cells, 19,155 genes; 48,244 with cell-type labels
- Melanoma cell line Drop-seq data with matched RNA FISH for 26 genes · n = 8,640 cells (8,498 after filtering); FISH on 7,000-88,000 cells per gene
- Methods & tools
- ALRA, MAGIC, SAVER, rsvd v0.9 (R), Seurat, FIt-SNE, ranger (random forest, R), STAR, HTSeq
- Reagents & environment
- R implementation using rsvd version 0.9 with q = 10 power iterations for the approximation and q = 2 for the 100 singular values used to pick k; ranger with default settings; k = 20 for the blood data and k = 48 for cortex.
- Assay conditions & compute
- Runtime comparisons were run on a single core with default parameters, SAVER with do.fast enabled; ALRA timings include normalisation, singular-value computation and completion. The authors note the randomised SVD step parallelises.
- Reproducibility gaps
- The R implementation and the analysis code (named ALRA-paper) are said to be freely available, but the addresses were lost in text extraction, so no link can be given. The purified blood matrices are described as coming from the 10x Genomics website, again without a usable link.
Limitations (5)
- Imputation is intentionally incomplete: where the reconstructed values of expressing and non-expressing cells overlap, as for CD4 and CD8A, real dropouts are left at zero. · stated by the authors
- The method assumes the true expression matrix is low-rank and non-negative; the symmetry of the error distribution around zero is stated as a hypothesis of independent theoretical interest, not proved. · stated by the authors
- The rank k is set by a heuristic on singular-value spacings, which the authors call effective but not derived; it is the method's one free parameter. · stated by the authors
- The main ground-truth evaluation is a multinomial simulation from nine bulk profiles, which produces data matching the low-rank, cell-type-block structure the method assumes. · noted in this summary
- Comparisons are partly indirect: MAGIC is dropped from two analyses on the strength of a published comparison, SAVER could not be run on the full cortex matrix, and no competitor's classifier error on that matrix is reported. · noted in this summary
Who should read:Read this if you impute dropouts in scRNA-seq matrices and want zeros that stay zero, or need an imputation step built to scale to very large datasets.
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