Dimensionality Reduction

A linear transformation that identifies the orthogonal axes of maximum variance in the data. PCA computes the eigenvectors of the covariance matrix to project cells onto principal components (PCs).

• Key property: Preserves global structure and distances
• Output: A ranked set of components where PC1 captures the most variance
• Use case: Initial dimensionality reduction before clustering, identifying batch effects, and visualizing the dominant sources of transcriptional variation
• Limitation: Assumes linear relationships; can obscure rare cell populations in the presence of strong batch effects

A non-linear technique that converts pairwise Euclidean distances into conditional probabilities representing similarities. It minimizes the Kullback-Leibler divergence between high-dimensional and low-dimensional probability distributions.

• Key property: Exceptional at preserving local neighborhoods and revealing fine-grained clusters
• Perplexity: A critical hyperparameter controlling the effective number of neighbors; typically tuned between 5 and 50
• Use case: Visualizing cellular subpopulations and validating cluster assignments in scRNA-seq data
• Limitation: Stochastic nature means plots are non-deterministic; global distances between clusters are meaningless; computationally intensive on large datasets

A manifold learning technique grounded in Riemannian geometry and algebraic topology. UMAP constructs a fuzzy topological representation of the high-dimensional data and optimizes a low-dimensional embedding to be as structurally similar as possible using cross-entropy.

• Key property: Balances local and global structure preservation better than t-SNE
• Speed: Significantly faster than t-SNE, scaling to millions of cells
• Parameters: n_neighbors controls local vs. global balance; min_dist controls embedding compactness
• Use case: The current standard for exploratory visualization of single-cell and multi-omics datasets; preserving trajectory continuity in developmental biology

A spectral method that embeds data by modeling a random walk (diffusion process) on the data graph. The Euclidean distance in the diffusion space approximates the diffusion distance, a robust measure of connectivity that accounts for all paths between points.

• Key property: Highly robust to noise and sampling artifacts; naturally captures branching trajectories
• Output: Diffusion components ordered by the timescale of the diffusion process
• Use case: Inferring developmental trajectories and pseudotime in single-cell data where continuous branching processes are expected
• Relation to PCA: Reduces to PCA when using a specific kernel and scaling; can be seen as a non-linear generalization

Neural network architectures consisting of an encoder that compresses input data into a low-dimensional bottleneck (latent space) and a decoder that reconstructs the original input. The latent space serves as a non-linear dimensionality reduction.

• Variational Autoencoders (VAEs): Enforce a probabilistic prior (typically Gaussian) on the latent space, enabling generative capabilities and smooth interpolation
• Key advantage: Can learn highly complex, non-linear manifolds that spectral methods miss
• Use case: Integrating multiple omics modalities into a shared latent space (e.g., scVI, totalVI); imputing missing data modalities; correcting batch effects within the latent representation
• Limitation: Requires careful hyperparameter tuning and significant computational resources for training

A linear technique that decomposes a multivariate signal into additive, statistically independent non-Gaussian components. Unlike PCA which decorrelates signals, ICA minimizes higher-order statistical dependencies.

• Key property: Recovers the underlying source signals of a mixed observation, assuming non-Gaussianity
• Use case: Deconvolving mixed cell-type signals from bulk tissue expression; identifying latent biological processes or pathways that act independently; removing technical artifacts that are statistically independent from biological signal
• Contrast with PCA: PCA finds orthogonal directions of maximum variance; ICA finds directions of maximum independence, often yielding more interpretable components for biological source separation

A linear transformation technique that identifies orthogonal axes of maximum variance in the data. In single-cell genomics, PCA is the foundational step for denoising and compressing the transcriptomic space before applying non-linear methods.

• Key property: Captures global structure; preserves large pairwise distances
• Limitation: Fails to preserve local neighborhood relationships in highly non-linear manifolds
• Typical usage: First 30-50 PCs retained as input for t-SNE or UMAP
• Mathematical basis: Eigendecomposition of the covariance matrix

A non-linear dimensionality reduction algorithm that converts pairwise Euclidean distances into conditional probabilities representing similarities. It excels at preserving local structure by modeling the probability distribution of neighbors in both high-dimensional and low-dimensional spaces.

• Perplexity: A hyperparameter that balances attention between local and global aspects, typically set between 5 and 50
• Crowding problem: Solved by using a heavy-tailed Student's t-distribution in the low-dimensional space
• Key caveat: Cluster sizes and inter-cluster distances are not interpretable; global topology is sacrificed for local fidelity

A manifold learning technique that constructs a fuzzy topological representation of the high-dimensional data and optimizes a low-dimensional embedding to be as structurally similar as possible. UMAP better preserves both local and global structure compared to t-SNE.

• Speed: Significantly faster than t-SNE, scaling to millions of cells
• Key parameters: n_neighbors controls local vs. global balance; min_dist controls point packing density
• Mathematical foundation: Based on Riemannian geometry and algebraic topology
• Advantage: Preserves more of the global data structure, including the continuity of developmental trajectories

A spectral method that embeds data by modeling a random walk over the data manifold. The Euclidean distance in the diffusion space approximates the diffusion distance—a robust measure of connectivity that accounts for all paths between points.

• Key strength: Highly robust to noise and sampling density variations
• Diffusion time parameter: Controls the scale of the manifold structure captured
• Application: Particularly effective for inferring pseudotemporal trajectories in single-cell data, as the diffusion process naturally follows developmental paths
• Relation to PCA: Reduces to PCA when using a linear kernel

A neural network architecture where an encoder compresses input data into a low-dimensional bottleneck representation, and a decoder reconstructs the original input. The bottleneck layer serves as a learned, non-linear dimensionality reduction.

• Variational Autoencoders (VAEs): Enforce a probabilistic latent space, enabling generative capabilities and integration of multiple omics modalities
• Denoising Autoencoders: Trained to reconstruct clean data from corrupted input, learning robust representations
• Advantage over PCA: Can capture complex non-linear relationships and be jointly optimized for downstream tasks
• Single-cell application: scVI and similar models use VAEs for batch correction and imputation simultaneously

Two distinct paradigms for reducing dimensionality. Feature selection identifies a subset of the original features (e.g., highly variable genes) and discards the rest, preserving interpretability. Feature extraction creates new synthetic features (e.g., principal components) that are combinations of the originals.

• Feature selection methods: Highly variable gene selection, mutual information, LASSO regularization
• Feature extraction methods: PCA, NMF, autoencoders
• Trade-off: Selection maintains biological interpretability; extraction often yields better downstream performance
• Best practice: Use feature selection to remove noise, then feature extraction for visualization