On Laplacian Eigenmaps for Dimensionality Reduction
summary
Claim: eigenfunctions of the Laplace-Beltrami operator give a low‑dimensional embedding; algorithm: build a similarity graph with Gaussian or k‑NN weights, form D − W graph Laplacian, solve the generalized eigenproblem, drop the zero eigenvector and use the next m eigenvectors as coordinates. Heat‑kernel analysis justifies the exponential weights (approximate Laplace‑Beltrami). Implements in scikit‑learn and unrolls S‑curve/sphere; sensitive to k and t and becomes costly past medium sizes (≈10k points).
AI-written summary · about
chapters (16)
- 0:00Laplacian Eigenmaps intro
- 1:10Motivating spectral geometry
- 4:42Graph Laplacian construction
- 9:23Solving eigenvalue problem
- 11:30Choosing graph weights
- 15:53Why graph Laplacian works
- 16:36Laplacian on manifolds
- 21:02Connecting graph and manifold
- 22:32Heat kernel role
- 26:30Theoretical justification
- 27:13scikit-learn implementation
- 29:32Spectral vs PCA embedding
- 30:40Neighborhood size impact
- 31:22Sphere with holes embedding
- 32:52References
- 33:41Q&A
related talks
53mConnection Laplacian, Hodge Laplacian, and Tensor Laplacian of a Graph
37mA Bluffer's Guide to Dimension Reduction
49mSpectral Approaches to Nearest Neighbor Search
40mTopological Techniques for Unsupervised Learning
12mDeep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS
01h 03mRecent Progress in High-Dimensional Learning
01h 34mAlgorithmic High-Dimensional Geometry II
34mDimensionally reducing data squeezing out the good stuff