MML-15-09: UMAP Mathematics: Fuzzy Simplicial Sets, Riemannian Manifolds & Cross-Entropy Objective – Practice Quiz

Master Uniform Manifold Approximation and Projection (UMAP) from first principles: local Riemannian metric normalization, nearest-neighbor distance offset $\rho_i$ for local connectivity, fuzzy simplicial set membership $p_{j|i}$, probabilistic t-conorm (fuzzy union) symmetrization $p_{ij} = p_{i|j} + p_{j|i} - p_{i|j}p_{j|i}$, smooth low-dimensional curve fitting $q_{ij} = (1 + a\|\mathbf{y}_i - \mathbf{y}_j\|^{2b})^{-1}$, and the Fuzzy Set Cross-Entropy objective. Aligns with Prerequisites: QS 15.8; Cognitive Level: Tier 2–4.