Chapter 5: Dimensionality Reduction: PCA, t-SNE & UMAP Geometry

Chapter 5: Dimensionality Reduction: PCA, t-SNE & UMAP Geometry

A single-cell dataset with 20,000 cells and 3,000 highly variable genes represents a 3,000-dimensional hypercube. Due to the "curse of dimensionality", Euclidean distance metrics lose discriminatory power in high dimensions. Dimensionality reduction compresses data into low-dimensional representations while retaining essential biological variance and topological structure.

1. Principal Component Analysis (PCA)

PCA computes orthogonal linear combinations of highly variable genes that maximize captured variance across cells.

# Seurat PCA workflow
pbmc <- RunPCA(pbmc, features = VariableFeatures(object = pbmc), npcs = 50)

# Inspect top genes driving PC1 and PC2
VizDimLoadings(pbmc, dims = 1:2, reduction = "pca")

# Plot PCA scatter
DimPlot(pbmc, reduction = "pca", group.by = "orig.ident")

2. Determining the Number of Significant PCs

Choosing too few PCs discards subtle biological subpopulations (e.g. dendritic cells); choosing too many includes technical noise. Two standard methods help determine the cutoff:

1. Elbow Plot: Plots the standard deviation of each PC. Look for the "elbow" point where the curve flattens out (typically around PC 20–30).
2. JackStraw Resampling: Permutes a subset of data (1%) and recalculates PCA to assign empirical \(p\)-values to PC loadings.

# Generate Elbow Plot
ElbowPlot(pbmc, ndims = 40)

3. t-SNE vs. UMAP: Algorithmic Architecture

While both algorithms project high-dimensional data into 2D, their mathematical formulations differ substantially:

• t-Distributed Stochastic Neighbor Embedding (t-SNE): Models pairwise Euclidean distances as probabilities under a Gaussian distribution in high dimensions and a Student-t distribution in low dimensions. It minimizes Kullback-Leibler (KL) divergence. t-SNE excels at preserving local neighborhoods but distorts global inter-cluster relationships and scales poorly to large cell numbers.
• Uniform Manifold Approximation and Projection (UMAP): Grounded in Riemannian geometry and algebraic topology. UMAP assumes data lies on a local Riemannian manifold with a fuzzy simplicial set. It preserves both local neighborhoods and broader global continuity, executes significantly faster, and scales efficiently to millions of cells.

4. Executing UMAP and Hyperparameter Tuning

# Seurat UMAP execution
pbmc <- RunUMAP(
    pbmc, 
    dims = 1:30, 
    n.neighbors = 30,    # Number of nearest neighbors (controls local vs global focus)
    min.dist = 0.3,       # Minimum distance between points in low-dim space (controls compactness)
    metric = "cosine"     # Distance metric
)

DimPlot(pbmc, reduction = "umap", label = TRUE)

# Scanpy PCA and UMAP workflow in Python
sc.tl.pca(adata, svd_solver='arpack', n_comps=50)
sc.pl.pca_variance_ratio(adata, log=True)

# Build k-NN neighborhood graph in PC space
sc.pp.neighbors(adata, n_neighbors=25, n_pcs=30, metric='cosine')

# Compute UMAP embedding
sc.tl.umap(adata, min_dist=0.3, spread=1.0)
sc.pl.umap(adata, color=['CD3D', 'MS4A1', 'CD14', 'NKG7'])