Linear Gene Programmes
======================


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   <div class="table-container">
     <table id="methods-table" class="display" style="width:100%">
       <thead>
         <tr>
           <th></th><th>Method</th><th>Year</th><th>Task</th>
           <th>Model</th><th>Published</th><th>Code</th>
         </tr>
       </thead>
       <tbody>
         <tr data-description="AMICI uses multi-headed cross-attention to model how receiver cells integrate information from spatial neighbors by generating query embeddings from learned cell-type-specific parameters that represent receiver identity, while neighboring non-self cells gene expression profiles generate keys and values through neural network embeddings. Attention scores are computed as distance-weighted softmax over query-key products, where each head learns interaction length scales and sender phenotypes, with the model reconstructing masked receiver expression as a residual from cell-type mean using attention-weighted aggregations of neighbor values. Sparsity-inducing penalties on attention weights and value matrices ensure only biologically relevant non-self neighbors contribute.">
           <td class="details-control"></td>
           <td><a href="https://www.biorxiv.org/content/10.1101/2025.09.22.677860v1.full.pdf">AMICI</a></td>
           <td>2025</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Multi-component Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">Cross-attention</span><span class="badge model-badge">Spatially-informed</span></div></td>


           <td class="published"><span style="color: #DC143C;">✗</span></td>
            <td><a href="https://github.com/azizilab/amici" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="CellCap is a deep generative model that extends CPA by incorporating cross-attention mechanisms between cell state and perturbation response (i.e., its basal latent space and the perturbation design matrix). Further, CellCap uses a variational autoencoder (VAE) framework with a linear decoder to identify sparse and interpretable latent factors.">
           <td class="details-control"></td>
           <td><a href="https://www.cell.com/cell-systems/fulltext/S2405-4712(25)00078-X">CellCap</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Multi-component Disentanglement</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Attention</span><span class="badge model-badge">Linear Decoder</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/broadinstitute/CellCap" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A family of contrastive latent variable models (cLVMs), where case data are modeled as the sum of background and salient latent embeddings, while control data are reconstructed solely from background embeddings: - cLVM with Gaussian likelihoods and priors - Sparse cLVM with horseshoe prior used to regularize the weights - Robust cLVM with a Student&#39;s t distribution - cLVM with automatic relevance determination to regularise the columns of the weight matrix - contrastive VAE, as a non-linear extension of the framework The shared concept across these models is that each model learns a shared set of latent variables for the background and target data, while salient latent variables are learnt solely for the target data.">
           <td class="details-control"></td>
           <td><a href="https://cdn.aaai.org/ojs/4414/4414-13-7453-1-10-20190706.pdf">cLVM</a></td>
           <td>2019</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Factor Models</span><span class="badge model-badge">Contastive VAE</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/kseverso/contrastive-LVM" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A modified version of PCA, where the covariance matrix (COV) is the difference between COV(case/perturbed) and αCOV(control/background). The hyperparameter α is used to balance having a high case variance and a low control variance. To provide some intuition, when α is 0, the model reduces to classic PCA on the case data. Optimal alphas (equal to k clusters) are identified using spectral clustering over a range of cPCA runs with different alphas, with selection based on the similarity of cPCA outputs.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41467-018-04608-8#Sec7">cPCA</a></td>
           <td>2018</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Modified PCA</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/abidlabs/contrastive" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A family of contrastive Poisson latent variable models (CPLVMs), based on a Gamma-Poisson hierarchical generative process: - CPLVM: The variational posterior is approximated using log-normal distributions, preserving non-negativity in the latent factors. - CGLVM: Extends CPLVM by allowing latent factors to take negative values, replacing Gamma priors with Gaussian priors and using a log-link function for the Poisson rates. Variational posteriors are modeled as multivariate Gaussians. The authors also propose a hypothesis testing framework, in which log-(ELBO)-Bayes is calculated between a Null model, omitting the salient latent space, and the full contrastive model. This framework is used to quantify global (across all genes) and joint expression changes in subsets of genes (akin to gene set enrichment analysis).">
           <td class="details-control"></td>
           <td><a href="https://projecteuclid.org/journals/annals-of-applied-statistics/volume-16/issue-3/Contrastive-latent-variable-modeling-with-application-to-case-control-sequencing/10.1214/21-AOAS1534.short">CPLVMs</a></td>
           <td>2022</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">NB likelihood</span><span class="badge model-badge">Factor Models</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/andrewcharlesjones/cplvm" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A non-negative matrix factorisation that decomposes gene expression matrices into common and condition-specific patterns. For each condition, the observed expression matrix is approximated as the sum of a common component - represented by a common feature matrix with condition-specific coefficient matrices - and a specific component unique to each condition, represented by its own feature matrix  and coefficients. The model uses an alternating approach to minimize the combined reconstruction error (squared Frobenius norm) across common and shared components.">
           <td class="details-control"></td>
           <td><a href="https://academic.oup.com/nar/article/47/13/6606/5512984">CSMF</a></td>
           <td>2019</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">NMF</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://www.zhanglab-amss.org/homepage/software.html" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Decipher is a hierarchical deep generative model that integrates and visualises single-cell RNA-seq data from normal and perturbed conditions, identifying shared and disrupted cell-state trajectories. Its architecture includes dual latent spaces: a low-dimensional state for detailed cell-state modeling and a two-dimensional space for visualisation-connected to gene expression through linear or single-layer neural network transformations. The model aligns trajectories by maintaining shared transcriptional programs for common biological processes across conditions.">
           <td class="details-control"></td>
           <td><a href="https://www.biorxiv.org/content/10.1101/2023.11.11.566719v2.full">Decipher</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Unsupervised Disentanglement</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Linear Decoder</span></div></td>


           <td class="published"><span style="color: #DC143C;">✗</span></td>
            <td><a href="https://github.com/azizilab/decipher" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="DIALOGUE identifies shared multicellular patterns across cell types and samples. It first constructs cell-type–specific data matrices by averaging features (e.g., gene expression or PCs) over samples or spatial niches. Then it applies multi-factor sparse canonical correlation analysis (referred to as penalized matrix decomposition (PMD)) to derive latent feature matrices that maximize cross-cell-type correlations under LASSO constraints. Following this initial PMD step, DIALOGUE employs correlation coefficients and permutation tests to determine which cell types contribute to each multicellular progarmmes (MCP). It then re-applies the PMD procedure in both a multi-way and a pairwise fashion, incorporating programs unique to the pairwise analysis into the downstream modeling. Finally, gene associated with MCPs are first identified using partial Spearman correlation and then refined through hierarchical mixed-effects modeling with covariate control.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41587-022-01288-0">DIALOGUE</a></td>
           <td>2022</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Sparse CCA</span><span class="badge model-badge">Partial Correlations</span><span class="badge model-badge">Mixed Linear Model</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/livnatje/DIALOGUE" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="EXPORT builds on the VEGA architecture by adding an auxiliary decoder that functions as an ordinal regressor, with an additional cumulative link loss to explicitly model dose-dependent response. ">
           <td class="details-control"></td>
           <td><a href="https://openreview.net/forum?id=f4nMJPKMkQ&referrer=%5Bthe%20profile%20of%20Xiaoning%20Qian%5D(%2Fprofile%3Fid%3D~Xiaoning_Qian1)">EXPORT</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/namini94/EXPORT" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="GEDI learns a shared latent space and, for each sample, estimates a specific reconstruction function that maps latent states to observed gene expression profiles. This design captures inter-sample variability and enables differential expression analysis along continuous cell-state gradients without relying on predefined clusters. Optionally, it can incorporate prior knowledge.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41467-024-50963-0?fromPaywallRec=false">GEDI</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Probabilistic</span><span class="badge model-badge">Sample-specific Decoders</span><span class="badge model-badge">PK Representations (optional)</span><span class="badge model-badge">RNA Velocity (optional)</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/csglab/GEDI" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
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         </tr>
         <tr data-description="GSFA is a two-layer, guided Factor Analysis model that quantifies the effects of genetic perturbations on latent factors. The model first factorizes the expression matrix Y into a factor matrix Z (normal prior) and gene loadings W (normal-mixture prior). Then, it captures the effect (β) of perturbation on factors using multivariate linear regression. Spike-and-slab prior is used to enforce sparsity on β, which can also analogously be seen as a causal graph. The linearity of GSFA further enables perturbation-associated, differentially-expressed genes to be identified. GSFA uses Gibbs sampling for inference.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41592-023-02017-4">GSFA</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Multi-component Disentanglement</span><span class="badge task-badge">Causal Structure</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Factor Analysis</span><span class="badge model-badge">Probabilistic</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/xinhe-lab/GSFA" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="MEFISTO extends MOFA+ by incorporating continuous covariates, such as time or spatial coordinates. While MOFA+ assumes that samples are independent and identically distributed, MEFISTO overcomes this limitation by combining the factor analysis framework with Gaussian Processes, which explicitly model the spatio-temporal relationships between samples (cells). MEFISTO can also align samples from different groups (e.g., individuals or species) that have misaligned temporal or spatial profiles.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41592-021-01343-9">MEFISTO</a></td>
           <td>2022</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Unsupervised Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Group Factor Model</span><span class="badge model-badge">Multi-modal</span><span class="badge model-badge">Spatially-informed</span><span class="badge model-badge">Time-resolved</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/bioFAM/MOFA2" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Memento is a differential expression framework that uses method-of-moments estimators under a multivariate hypergeometric model, where a gene’s mean is derived from Good-Turing corrected counts scaled by total cell counts. Differential variability is quantified as the variance remaining after accounting for mean-dependent effects (residual variance), while the covariance (pairwise association) between genes is estimated from the off-diagonal elements of the resulting variance-covariance matrix. Efficient permutation is achieved through a bootstrapping strategy that leverages the sparsity of unique transcript counts.">
           <td class="details-control"></td>
           <td><a href="https://www.cell.com/cell/fulltext/S0092-8674(24)01144-9">Memento</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Differential Analysis</span><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Feature Relationships</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Hypergeometric test</span><span class="badge model-badge">Probabilistic</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/yelabucsf/scrna-parameter-estimation" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Mixscape aims to classify CRISPR-targeted cells into perturbed and not perturbed (escaping). To eachive that, Mixscape computes a local perturbation signature by subtracting each cell’s mRNA expression from the average of its k nearest NT (non-targeted) control neighbors. Differential expression testing between targeted and NT cells then identifies a set of DEGs that capture the perturbation response. These DEGs are used to define a perturbation vector-essentially, the average difference in expression between targeted and NT cells, which projects each cell’s DEG expression onto a single perturbation score. The Gaussian mixture model is applied to these perturbation scores, with one component fixed to match the NT distribution, while the other represents the perturbation effect. This model assigns probabilities that classify each targeted cell as either perturbed or escaping. Additionally, the authors propose visualisation with Linear Discriminant Analysis and UMAP, aiming to identify a low-dimensional subspace that maximally discriminates the mixscape-derived classes.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41588-021-00778-2#Sec11">Mixscape</a></td>
           <td>2021</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Perturbation Responsiveness</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Gaussian Mixture Model</span><span class="badge model-badge">LDA</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/satijalab/seurat" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
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         </tr>
         <tr data-description="Multi-Omics Factor Analysis v2 (MOFA+) builds builds upon Bayesian Group Factor Analysis, using a matrix factorization approach to infer a low-dimensional representation of typically multi-modal data in the form of latent factors that capture the primary sources of variation across different data types (e.g., RNA expression, DNA methylation) and sample groups. MOFA+ is trained using GPU-accelerated stochastic variational inference which ensures scalability. It also combines Automatic Relevance Determination (ARD) priors, which disentangle variation shared across or unique to specific data modalities and sample groups, with sparsity-inducing spike-and-slab priors on both factors and weights.">
           <td class="details-control"></td>
           <td><a href="https://link.springer.com/article/10.1186/s13059-020-02015-1">MOFA+</a></td>
           <td>2020</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Unsupervised Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Group Factor Model</span><span class="badge model-badge">Multi-modal</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/bioFAM/MOFA2" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Multicellular factor analysis repurposes MOFA by treating pseudobulked cell types as views. Each patient is represented by multiple views - one per cell type - summarizing gene expression. MOFA+ ised then used to identify latent factors that capture coordinated variability across these views, with loadings indicating cell-type-specific gene contributions. ">
           <td class="details-control"></td>
           <td><a href="https://elifesciences.org/articles/93161">MOFAcell</a></td>
           <td>2023</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Group Factor Analysis (MOFA+)</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/saezlab/MOFAcellulaR" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="MUSIC evaluates sgRNA knockout efficiency and summarises perturbation effects using topic modeling. Following preprocessing steps, MUSIC removes low-efficiency (non-targeted) cells based on the cosine similarity of their differential expression genes, excluding perturbed cells with profiles more similar to controls. Next, highly dispersed DE genes are selected and their normalised expression values are used as to fit a topic model, where cells are treated as documents and gene counts as words. Topics are then ranked according to overall effect, their relevance to each perturbation, and perturbation similarities.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41467-019-10216-x">MUSIC</a></td>
           <td>2019</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Perturbation Responsiveness</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Topic Model</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/bm2-lab/MUSIC" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="MuVI is a multi-view factor analysis that encodes prior knowledge by imposing structured sparsity on view-specific factor loadings via a weighted, regularized horseshoe prior. Specifically, it uses a weight parameter that controls the variance of each loading; e.g., by default, it is set to 0.99 for genes known to belong to a gene set and 0.01 for genes which do not (are uknown). Using this hieararchical regulairisation strategy, MuVI directly associates latent factors with corresponding gene sets while still allowing for the de novo identification of additional genes relevant to a given factor.">
           <td class="details-control"></td>
           <td><a href="https://proceedings.mlr.press/v206/qoku23a.html">MuVi</a></td>
           <td>2023</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Unsupervised Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Group Factor Model</span><span class="badge model-badge">PK Representations</span><span class="badge model-badge">Multi-modal</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/MLO-lab/MuVI" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="NicheCompass, the spatial sucessor of ExpiMap, employs multiple decoders: one graph decoder reconstructs the spatial adjacency matrix via an adjacency loss to ensure that spatially-neighboring observations have similar latent representations, while separate (masked) decoders - one for each cell’s own features and one for its aggregated neighborhood features - reconstruct the omics data. By masking the data reconstruction according to prior knowledge, each latent variable is associated with a gene program (subclassified according inter- or  intracellular signalling). Additionally, it learns de novo gene programs that capture novel, spatially coherent expression patterns, not covered by the prior knowledge. By default, it replaces the Group lasso loss of Expimap with a a dropout mechanism to prune uninformative prior knowledge sets.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41588-025-02120-6">NicheCompass</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Graph VAE</span><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">NB Likelihood</span><span class="badge model-badge">Spatially-informed</span><span class="badge model-badge">PK Representations</span><span class="badge model-badge">Multi-modal</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/Lotfollahi-lab/nichecompass." class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="ontoVAE uses a multi-layer, linear decoder, structured to represent hierarchical prior knowledge - e.g. layers can represent gene ontology level.  To preserve connections beyond adjacent layers, the decoder concatenates outputs from previous layers with the current layer’s input, with binary masks ensuring that only valid parent–child and gene set relationships are captured. Decoder weights are constrained to be positive to preserve directional pathway activity, with each ontology term represented by three neurons whose average activation reflects its activity.">
           <td class="details-control"></td>
           <td><a href="https://academic.oup.com/bioinformatics/article/39/6/btad387/7199588">ontoVAE</a></td>
           <td>2023</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/hdsu-bioquant/onto-vae" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A probabilistic model that builds on cPCA, additionally proposing a case-control-ratio-adjusted α as a more interpretable alternative to the same parameter in cPCA (see comment above).">
           <td class="details-control"></td>
           <td><a href="https://projecteuclid.org/journals/annals-of-applied-statistics/volume-18/issue-3/Probabilistic-contrastive-dimension-reduction-for-case-control-study-data/10.1214/24-AOAS1877.short">PCPCA</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">modified PCA</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/andrewcharlesjones/pcpca" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="In pmVAE, each predefined pathway is modeled as a VAE that learns a (local) multidimensional latent embedding for the genes in that pathway. Each VAE module minimizes a size-weighted local reconstruction loss based solely on its pathway’s genes, while the (local) latent embeddings from all pathways are concatenated to form a global representation. ">
           <td class="details-control"></td>
           <td><a href="https://icml-compbio.github.io/2021/papers/WCBICML2021_paper_24.pdf">pmVAE</a></td>
           <td>2021</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Multiple VAEs</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/ratschlab/pmvae " class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="scETM uses a standard VAE encoder with a softmax layer to obtain a cell-by-topic matrix, paired with a linear decoder based on matrix tri-factorisation that reconstructs the data from the cell-by-topic matrix, along with topics-by-embedding α, and embedding-by-genes ρ matrices. This structure allows the latent topics to be directly interpreted as groups of co-expressed genes and can optionally integrate prior pathway (prior knowledge) information as a binary mask.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41467-021-25534-2">scETM</a></td>
           <td>2021</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Embedding Topic Model</span><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/hui2000ji/scETM" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A non-negative matrix factorisation method that decomposes single-cell gene expression data into common and condition-specific gene modulees. Each sample’s expression matrix is modeled as the sum of a shared component (W₂V) and condition-specific components (W₁Hⱼ), plus residual noise. The approach minimizes a loss function combining reconstruction error (Frobenius norm) with regularisation terms that control module scale and inter-condition similarity.">
           <td class="details-control"></td>
           <td><a href="https://link.springer.com/article/10.1186/s13059-022-02649-3#Sec11">scINSIGHT</a></td>
           <td>2022</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Contrastive Disentanglement</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">NMF</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/Vivianstats/scINSIGHT" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="scITD constructs a three-dimensional tensor (donors × genes × cell types) by generating donor-by-gene pseudobulk matrices for each cell type. Tucker decomposition then decomposes this tensor into separate factor matrices for donors, genes, and cell types, along with a core tensor that captures their interactions as latent multicellular expression patterns. The gene factors and core tensor are rearranged into a loading tensor analogous to PCA loadings, while the donor factor matrix represents sample scores. Finally, to improve interpretability, a two-step rotation is carried out - first applying ICA to the gene factors and then varimax to the donor factors.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41587-024-02411-z">scITD</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Tensor Decomposition (Tucker)</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/kharchenkolab/scITD" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="A sparse version of contrastive PCA that enhances interpretability in high-dimensional settings by integrating l1 regularisation into an iterative procedure to estimate sparse loadings and principal components">
           <td class="details-control"></td>
           <td><a href="https://academic.oup.com/bioinformatics/article/36/11/3422/5807607">scPCA</a></td>
           <td>2020</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Contrastive Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Modified PCA</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/PhilBoileau/EHDBDscPCA" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="scRank infers cell type-specific Gene Programmes from untreated scRNA-seq data by constructing co-expression networks via principal component regression with random subsampling and integrating them using tensor decomposition. It simulates drug perturbation by modifying the drug targets&#39; outgoing edges to generate an in-sillico perturbed network, and then aligns the untreated and perturbed networks via Laplacian eigen-decomposition. In this low-dimensional space, the distances between corresponding gene nodes quantify gene-level changes due to the perturbation. These distances, weighted by network connectivity (e.g., outgoing edge strength normalised by node degree) and extended through two-hop diffusion, are aggregated to yield a composite perturbation score that ranks cell types by their predicted drug responsiveness.">
           <td class="details-control"></td>
           <td><a href="https://www.cell.com/cell-reports-medicine/fulltext/S2666-3791(24)00260-X">scRANK</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Perturbation Responsiveness</span><span class="badge task-badge">GRN Inference</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">PC Regression</span><span class="badge model-badge">Tensor Decomposition (PARAFAC)</span><span class="badge model-badge">Network Diffusion</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/ZJUFanLab/scRank" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Spectra decomposes a gene expression matrix into cell-by-factor and factor-by-gene matrices, while integrating prior knowledge gene sets and cell-type labels. It explicitly models both global and cell-type–specific factors by incorporating cell-type labels, thereby disentagling the typically dominating cell-type variation from shared Gene Programmes. Gene sets are represented as a gene–gene knowledge graph, and a penalty term based on a weighted Bernoulli likelihood, guides the factorisation toward preserving this graph. Yet, it also permits the data-driven discovery of novel programs by &#39;detaching&#39; factors from the prior. Spectra can also include cell-type-specific prior knowledge gene sets (e.g. T cell antigen receptor activation programmes can be limited to T cells)">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41587-023-01940-3">Spectra</a></td>
           <td>2023</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Multi-component Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Poisson Likelihood</span><span class="badge model-badge">Factor Analysis</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/dpeerlab/spectra" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="STAMP is an interpretable, spatially-aware dimension reduction method that models the gene expression using Gamma Poisson likelihood, determined by a combination of latent topic proportions (logNormal) ,gene modules, and background residuals. The model employs a structured regularised horseshoe prior on the gene modules to induce sparsity. To incorporate spatial context, STAMP utilises a simplified graph convolutional network as its inference network. STAMP can be extended to handle multi-sample data by adding a batch correction term (via batch-specific loading matrices) and to analyse time-series data by allowing gene modules to vary over time via a Gaussian process prior.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41592-024-02463-8">STAMP</a></td>
           <td>2024</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span><span class="badge task-badge">Unsupervised Disentanglement</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Spatially-informed</span><span class="badge model-badge">Time-resolved</span><span class="badge model-badge">Graph encoder</span><span class="badge model-badge">Factor Analysis</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/JinmiaoChenLab/scTM" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="VEGA replaces conventional fully connected decoder with a sparse linear decoder that uses a binary gene membership mask, assingning latent variables to a pre-defined collection of gene sets.">
           <td class="details-control"></td>
           <td><a href="https://www.nature.com/articles/s41467-021-26017-0">VEGA</a></td>
           <td>2023</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">VAE</span><span class="badge model-badge">Linear Decoder</span><span class="badge model-badge">PK Representations</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/LucasESBS/vega/" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
         <tr data-description="Waddington-OT models developmental processes as time-varying  probability distributions in gene expression space and infers temporal couplings by solving an entropy-regularized, unbalanced optimal transport problem. Growth rate, estimated leveraging expression levels of genes associated with proliferation and apoptosis, is taken into consideration via unbalanced OT. Additionally, uses spectral clustering to obtain Gene Programmes, and subsequently associate those to predictive TFs.">
           <td class="details-control"></td>
           <td><a href="https://www.sciencedirect.com/science/article/pii/S009286741930039X?via%3Dihub">Waddington-OT</a></td>
           <td>2019</td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge task-badge">Trace Cell Populations</span><span class="badge task-badge">Linear Gene Programmes</span></div></td>

           <td><div style="display: flex; flex-wrap: wrap; gap: 0.3em;"><span class="badge model-badge">Unbalanced OT</span><span class="badge model-badge">Entropy-regularized Sinkhorn</span></div></td>


           <td class="published"><span style="color: #2E8B57;">✓</span></td>
            <td><a href="https://github.com/broadinstitute/wot" class="github-link">
                  <i class="fab fa-github" aria-hidden="true"></i>
                  <span class="sr-only">GitHub</span>
                </a></td>
         </tr>
       </tbody>
     </table>
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