Dimensionality reduction

Method	Linear	Local	Global	Scales well	Typical use
PCA	✓	✗	✓	✓✓✓	Baseline, preprocessing
t-SNE	✗	✓✓✓	✗	✗	excells in local
UMAP	✗	✓✓	✓	✓✓	General-purpose
TRIMAP	✗	✓✓	✓✓	✓✓	global triplets
PaCMAP	✗	✓✓	✓✓	✓✓✓	good alround
LocalMAP	✗	✓✓	✓✓	✓✓✓	good alround
LLE	✗	✓✓	✗	✗	...

PCA — Principal Component Analysis

Category: Linear
Preserves: Global variance
Scales to: Very large datasets

Summary

PCA projects the data onto orthogonal axes that maximize variance. It provides a linear approximation of the data and is often used as a baseline method or as a preprocessing step prior to nonlinear dimensionality reduction.

In k-mer ordination, PCA captures dominant compositional gradients but may fail to represent nonlinear relationships between samples.

Reference

Pearson, K. (1901)
On Lines and Planes of Closest Fit to Systems of Points in Space
Philosophical Magazine
DOI: 10.1080/14786440109462720

UMAP — Uniform Manifold Approximation and Projection

Category: Nonlinear, manifold learning
Preserves: Local structure, with limited global structure preservation
Scales to: Large datasets

Summary

UMAP constructs a fuzzy topological representation of the high-dimensional data and optimizes a low-dimensional embedding that preserves local neighborhood relationships. Compared to t-SNE, UMAP generally offers better runtime performance and improved preservation of global structure.

UMAP is well suited for large k-mer count matrices where local similarity between sequences is biologically meaningful.

Reference

McInnes, L., Healy, J., & Melville, J. (2018)
UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction
arXiv DOI: 10.48550/arXiv.1802.0342

t-SNE — t-distributed Stochastic Neighbor Embedding

Category: Nonlinear, probabilistic
Preserves: Local structure
Scales to: Small–medium datasets

Summary

t-SNE models pairwise similarities as probability distributions and minimizes their divergence between high- and low-dimensional spaces. It excels at revealing local clusters but does not preserve global distances.

t-SNE embeddings are sensitive to hyperparameters and random initialization.

Reference

van der Maaten, L., & Hinton, G. (2008)
Visualizing Data using t-SNE
Journal of Machine Learning Research
DOI: 10.48550/arXiv.2008.09237

TRIMAP

Category: Nonlinear
Preserves: Global structure with local constraints
Scales to: Medium–large datasets

Summary

TRIMAP constructs triplet constraints to preserve relative distances between points. Unlike t-SNE, TRIMAP explicitly balances local and global structure, making it suitable for exploratory analysis of large datasets.

Reference

Amid, E., & Warmuth, M. K. (2019)
TriMAP: Large-scale Dimensionality Reduction Using Triplets
arXiv
DOI: 10.48550/arXiv.1910.00204

PaCMAP — Pairwise Controlled Manifold Approximation

Category: Nonlinear
Preserves: Local, mid-range, and global structure
Scales to: Large datasets

Summary

PaCMAP explicitly optimizes three distance regimes: near, mid-range, and far. This allows it to preserve both cluster structure and global geometry more effectively than t-SNE or UMAP in many settings.

Reference

Wang, Y., Huang, H., Rudin, C., & Shaposhnik, Y. (2021)
Understanding How Dimension Reduction Tools Work
Journal of Machine Learning Research
DOI: 10.48550/arXiv.2012.04456

LocalMAP

Category: Nonlinear
Preserves: Strong local structure
Scales to: Medium datasets

Summary

LocalMAP is a variant of PaCMAP that emphasizes local neighborhood preservation. It is particularly useful when fine-scale structure is of primary interest.

Reference

Wang, Y., Huang, H., & Rudin, C. (2021)
PaCMAP and LocalMAP: Manifold Learning with Controlled Locality
arXiv
DOI: 10.48550/arXiv.2103.03167

LLE — Locally Linear Embedding

Category: Nonlinear
Preserves: Local linear structure
Scales to: Small datasets

Summary

LLE reconstructs each data point as a linear combination of its neighbors and finds a low-dimensional embedding that preserves these relationships.

LLE is sensitive to noise and neighborhood size and is generally unsuitable for very large k-mer datasets.

Reference

Roweis, S. T., & Saul, L. K. (2000)
Nonlinear Dimensionality Reduction by Locally Linear Embedding
Science
DOI: 10.1126/science.290.5500.2323