Week 06 — Learning on Manifolds

Molecular structures, brain scans, single-cell embeddings — data that lives on curved surfaces where Euclidean distance lies. Manifold-aware learning is the fix.

Lecture

Riemannian manifolds at a working level · the exponential and logarithmic maps · tangent-space methods · geodesic distance vs Euclidean · hyperbolic embeddings (Nickel-Kiela 2017).

Read before the lecture

• Nickel and Kiela, *Poincaré Embeddings for Learning Hierarchical Representations* (NeurIPS 2017)

Code lab

Hyperbolic embedding of a taxonomy

Embed WordNet’s mammal subtree in 2-D Euclidean and 2-D Poincaré-ball space. Compare nearest-neighbor structure.

Notebook: lab02-hyperbolic.ipynb  ·  Dataset: WordNet (NLTK).

Reference text for this week: chapter 06 of the bilingual notes — EN PDF · FR PDF.