Dimensionality Reduction and Manifold Estimation
			   PMF-Mathematics
			     Winter, 2022

			 Project Suggestions

These projects should be done individually, without collaboration.
Prepare a talk with approximately 20 slides, expecting to take 30
minutes plus 15 minutes for questions.

NOTE: The references are not exhaustive: you are welcome to find other
sources.

1. Prove some version of the Whitney Embedding Theorem, for smooth
noncompact manifolds, or for compact differentiable manifolds.
Discuss the technical issues and the important differences in the
various versions.  

References:
https://www.math.wustl.edu/~victor/classes/pmf/WOMPtalk-Manifolds.pdf
https://www.math.wustl.edu/~victor/classes/pmf/PartOfUnityLocFiniteRefinements.pdf
https://www.math.wustl.edu/~victor/classes/pmf/Brahim-Abdenbi-presentation-Whitney.pdf
https://www.math.wustl.edu/~victor/classes/pmf/WhitEmb-Lec09.pdf
https://www.math.wustl.edu/~victor/classes/pmf/lecture10-WhitEmbed.pdf


2. Prove the Birkhoff Ergodic Theorem.  Prove the Von Neumann Ergodic
Theorem.  Discuss their differences and compare them with the mean and
maximal ergodic theorems.  Try to find hypotheses that guarantee a
given rate of convergence, and find examples with particularly slow
convergence.

References:
https://www.math.wustl.edu/~victor/classes/pmf/Yunis.pdf
https://www.math.wustl.edu/~victor/classes/pmf/MetropHastingsEtc.pdf


3. Find coordinates in 2 dimensions for the full 2000-point Swiss Roll
data set using graph Laplacian eigenvectors (for an undirected
unweighted adjacency matrix with some reasonable choice of edges) and
diffusion maps (for a weighted similarity matrix with some choice of
sigma).  Experiment with parameters and discuss how this changes the
results.  

References:
https://www.math.wustl.edu/~victor/classes/pmf/sr2000x3.dat
https://www.math.wustl.edu/~victor/classes/pmf/06swiss.txt
https://www.math.wustl.edu/~victor/classes/pmf/ch1-EigGraphLapl.pdf
https://www.math.wustl.edu/~victor/classes/pmf/GraphLaplacian-tutorial.pdf
https://www.math.wustl.edu/~victor/classes/pmf/Lafon06.pdf
https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf