Canonical Correlation Analysis (CCA)

A statistical framework for the unsupervised integration of multiple omics data types that infers a low-dimensional set of latent factors capturing the principal sources of variation across different data modalities from the same set of samples.

• Unlike CCA, which finds maximally correlated projections between two datasets, MOFA generalizes to three or more data modalities simultaneously
• Decomposes variation into factors shared across all omics layers and factors private to individual data types
• Provides interpretable latent representations that can be correlated with clinical covariates or outcomes
• Built on a Bayesian probabilistic model with sparsity-inducing priors for feature selection
• Widely used for identifying coordinated multi-omics signatures in cancer subtyping and drug response studies

An algorithm that integrates multiple omics data types by constructing a sample-similarity network for each data type and then iteratively fusing them into a single consensus network that captures shared and complementary information.

• Operates on the principle that true biological signals manifest consistently across multiple data types, while noise is modality-specific
• Constructs a patient-by-patient similarity matrix for each omics layer (e.g., mRNA expression, DNA methylation, miRNA)
• Iteratively updates each network by diffusing information from the others using a nonlinear message-passing approach
• Converges to a fused network where clusters represent clinically meaningful subtypes
• Particularly effective for patient stratification in heterogeneous diseases like glioblastoma and ovarian cancer
• Unlike CCA's linear correlation maximization, SNF captures nonlinear relationships through graph-based fusion

A self-supervised learning paradigm that trains a model to pull representations of similar data points together and push dissimilar ones apart in an embedding space, used in multi-omics to align cells or features across different modalities without paired labels.

• Forms the backbone of modern single-cell foundation models like scGPT and Geneformer
• In multi-omics integration, contrastive objectives can align scRNA-seq and scATAC-seq profiles from the same cell type without requiring paired measurements
• The InfoNCE loss maximizes mutual information between positive pairs while minimizing it for negative pairs
• Enables zero-shot transfer learning across modalities by learning a shared latent space
• Extends CCA's linear correlation framework to nonlinear deep learning architectures with explicit alignment objectives
• Key variants include SimCLR, MoCo, and CLIP-style multimodal contrastive learning adapted for biological data

A mathematical framework for finding the most efficient mapping between two probability distributions, applied in single-cell biology to align cells from different batches, time points, or omics modalities by minimizing a cost function based on their feature similarity.

• Formulated through the Kantorovich relaxation, which allows probabilistic, fractional assignments rather than strict one-to-one mappings
• In multi-omics, optimal transport can align scRNA-seq cells with scATAC-seq cells by treating their latent representations as distributions to be matched
• The Gromov-Wasserstein distance extends optimal transport to align datasets where features are not directly comparable, preserving intra-domain distances
• Provides a principled alternative to CCA when the relationship between modalities is nonlinear or when feature correspondences are unknown
• Used in tools like moscot and Pamona for spatiotemporal trajectory alignment and cross-modality mapping
• The entropic regularization parameter controls the trade-off between precise alignment and computational tractability

A generative deep learning architecture that learns a probabilistic, compressed latent representation of input data, widely adapted in single-cell biology for integrating multi-omics data, imputing missing modalities, and modeling complex cellular distributions.

• Encodes high-dimensional omics data into a low-dimensional Gaussian latent space through an encoder network and reconstructs it via a decoder
• The KL divergence regularization term ensures the latent space is smooth and continuous, enabling interpolation between cell states
• Multi-modal VAEs like totalVI and scVI jointly model RNA and protein measurements from CITE-seq experiments
• Extends CCA's linear projections to nonlinear manifold learning, capturing complex gene expression programs
• Enables in silico perturbation by traversing the latent space to predict unseen cellular states
• The generative nature allows for uncertainty quantification and hypothesis testing on latent representations

A mathematical technique for transforming high-dimensional omics data into a lower-dimensional space for visualization and noise reduction, with algorithms like PCA, t-SNE, and UMAP being essential for exploring cellular heterogeneity.

• Principal Component Analysis (PCA) performs linear dimensionality reduction by finding orthogonal axes of maximum variance, serving as the foundation for CCA's correlation-maximizing projections
• t-SNE preserves local neighborhood structure through probabilistic pairwise similarities, excelling at revealing discrete cell clusters
• UMAP balances local and global structure using a topological framework based on Riemannian geometry and fuzzy simplicial sets
• CCA can be viewed as a supervised dimensionality reduction technique where the objective is to find projections that maximize cross-dataset correlation rather than within-dataset variance
• Modern single-cell analysis pipelines typically use PCA for initial denoising before applying nonlinear methods for visualization
• The choice of dimensionality reduction method significantly impacts downstream clustering, trajectory inference, and integration results