LIGER (Linked Inference of Genomic Experimental Relationships)

LIGER's core algorithm extends traditional NMF to a multi-dataset context. It decomposes each dataset's expression matrix into a set of shared metagenes (factors common across all batches) and dataset-specific metagenes (factors capturing unique technical or biological variation). This joint factorization is solved using block coordinate descent, alternating between updating the shared factor matrix, the dataset-specific factor matrices, and the cell loading matrices. The non-negativity constraint ensures that factors represent additive combinations of gene expression programs, yielding highly interpretable components that correspond to discrete cell types or biological processes.

After iNMF decomposition, LIGER constructs a joint cell neighborhood graph using the shared factor loadings (the H matrix). Because these loadings represent each cell's expression in terms of the common metagenes, they are directly comparable across batches. LIGER builds a k-nearest neighbor graph in this shared factor space, effectively aligning cells from different batches that exhibit similar biological states. This graph is then used for downstream tasks such as clustering, trajectory inference, and differential expression analysis, all performed on the batch-corrected, integrated representation.

To further refine the integration, LIGER applies a quantile normalization step to the shared factor loadings. This process forces the empirical distribution of each factor's loadings to be identical across all datasets. By matching quantiles, LIGER corrects for any remaining distributional differences in the factor space that may arise from varying sequencing depths or other technical artifacts. This step is particularly effective at aligning datasets with substantially different total cell counts or dynamic ranges, ensuring that a shared factor represents the same biological signal at the same magnitude in every batch.

LIGER includes an online iNMF algorithm designed to handle massive single-cell datasets that exceed available memory. Instead of loading all cells into RAM, the algorithm processes the data in mini-batches, iteratively updating the factor matrices using stochastic gradient descent. This approach allows LIGER to integrate millions of cells across dozens of batches on standard computational infrastructure. The online variant maintains the interpretability of the standard iNMF while dramatically reducing memory footprint, making it suitable for atlas-scale projects like the Human Cell Atlas.

A key diagnostic feature of LIGER is the explicit modeling of dataset-specific factors. These factors capture variation that is unique to a single batch and not shared with others. By examining the gene loadings of these factors, analysts can identify the molecular signatures of batch effects, such as ribosomal protein expression or mitochondrial read contamination. Alternatively, dataset-specific factors can reveal genuine biological signals present in only one condition, such as a disease-specific cell state. This decomposition provides a transparent view of what variation is being removed versus preserved during integration.

LIGER performs consensus clustering on the integrated shared factor space to identify cell types robustly. The algorithm runs multiple rounds of community detection on the shared factor neighborhood graph with varying resolution parameters, then aggregates the results to find stable cluster assignments. This approach mitigates the stochasticity inherent in single-resolution clustering and provides a confidence metric for each cell's assignment. The resulting clusters represent cell populations that are consistently identified across all integrated batches, free from batch-specific biases.

The core mathematical engine of LIGER. Unlike standard NMF, iNMF decomposes multiple datasets simultaneously into a set of shared metagenes (factors common across all batches) and dataset-specific metagenes (factors capturing unique technical or biological variation). This joint decomposition is solved via block coordinate descent, alternating between updating the shared factor matrix and each dataset's specific factor matrix while enforcing non-negativity constraints. The shared factors define a low-dimensional space where cells from different batches are directly comparable.

After iNMF decomposition, LIGER constructs a k-nearest neighbor graph using the shared factor loadings (the H matrix) rather than the original gene expression space. This graph connects cells across batches based on their shared biological identity, effectively ignoring dataset-specific noise. The graph is then used for joint clustering via Louvain or Leiden community detection and for generating Uniform Manifold Approximation and Projection (UMAP) visualizations. The key insight is that batch correction is achieved not by transforming the raw data, but by redefining the neighborhood relationships.

A post-hoc refinement step unique to LIGER's workflow. After joint clustering, LIGER applies quantile normalization within each identified cluster across batches. This step aligns the empirical expression distributions of genes for cells assigned to the same biological group but originating from different datasets. This within-cluster normalization corrects for residual technical variation in feature magnitudes that may persist after the factor-level alignment, ensuring that downstream differential expression tests are not biased by batch-specific scaling differences.

A quantitative metric to evaluate integration quality. The alignment score measures the proportion of a cell's k-nearest neighbors in the shared factor space that originate from a different batch than the query cell. A score near 1.0 indicates perfect mixing, where batch identity is irrelevant to local neighborhood composition. This metric complements Local Inverse Simpson's Index (LISI) and kBET by directly quantifying cross-batch connectivity in the reduced dimensional space. LIGER's alignment is often visualized using a Sankey diagram showing the flow of cells from original batches into integrated clusters.

A memory-efficient variant designed for massive datasets exceeding physical RAM limits. Online iNMF processes data in mini-batches, updating the shared factor matrix incrementally using stochastic gradient descent. This allows LIGER to integrate millions of cells without loading the entire dataset into memory simultaneously. The algorithm converges to a solution approximating the full batch iNMF, trading a small amount of precision for dramatic gains in scalability. This is critical for atlas-scale projects like the Human Cell Atlas.

An extension called Unshared Integrative NMF (UINMF) handles datasets that do not share the same feature space. For example, integrating scRNA-seq data with scATAC-seq or CITE-seq antibody-derived tags. UINMF learns a shared cell factor matrix from the intersecting features while simultaneously learning separate gene and peak factor matrices for the unshared modalities. This enables the joint analysis of transcriptomic and epigenomic profiles without requiring feature conversion or imputation, preserving the native structure of each data type.