What is the role of approximation techniques in nonlinear dimensionality reduction?

Approximation techniques make nonlinear DR scalable by reducing computational complexity.

Nonlinear methods like t-SNE and UMAP involve expensive pairwise computations. Approximation techniques such as Barnes-Hut trees, nearest neighbor graphs, and stochastic sampling reduce complexity from O(n²) to near O(n log n). These approximations trade exactness for scalability while preserving essential structure.