E31 Manifold learning and spectral methods

Why this course

Fills the gap between linear algebra, geometry and dimensionality reduction used in data science.

Prerequisites

Required

B08 Linear algebra II: abstract structure, B09 Real analysis I, B11 Probability I

Recommended

P04 Smooth manifolds

Can be taken at the same time

none

Everything required, transitively (10 courses)

B01, B02, B03, B04, B05, B08, B09, B11, F01, F02

Courses that require this one

E44

Concept-level units

Units follow the syllabus order, as a recommended sequence rather than a gate.

1. Principal components and the singular value decomposition as geometry
2. Graph Laplacians and spectral clustering
3. Isomap, locally linear embedding, Laplacian eigenmaps
4. Diffusion maps and the heat kernel
5. t-SNE and UMAP: what they preserve and what they distort
6. The manifold hypothesis and intrinsic dimension
7. Johnson-Lindenstrauss and random projections
8. Reading an embedding plot critically

Assessable outcome

Choose and justify a dimensionality-reduction method for a data set, and say what its picture can and cannot show.

Assessment design

• topic exercises
• mixed problem sets
• unseen proof or derivation problems
• cumulative examination

cumulative; uses spaced retrieval

Resources

Every title here is candidate, unverified unless marked verified. Titles were suggested while planning. Nobody has checked that they exist, what they cost or how they are licensed.

• Murphy · Probabilistic Machine Learning: An Introduction (free)
• Belkin and Niyogi · Laplacian eigenmaps (paper)
• van der Maaten and Hinton · Visualizing data using t-SNE (paper)
• Deisenroth, Faisal, Ong · Mathematics for Machine Learning (free)

Bridges and laboratories

No bridge or laboratory is attached to this course yet.

You and this course