NOTES, SLIDES, AND PAPERS

Date: 01-09-2025

Introduction: Slides

Date: 01-14-2025

"You can always recognize truth by its beauty and simplicity"―Richard P. Feynman 

• Singular Value Decomposition (SVD) 

• Example 1

Papers

1. Lieberman-Aiden, Erez, ..., Groudine Mark, ..., Lander Eric. "Comprehensive mapping of long-range interactions reveals folding principles of the human genome." Science 326.5950 (2009): 289-293. 

2. Turk, Matthew, and Alex Pentland. "Eigenfaces for recognition." Journal of cognitive neuroscience 3.1 (1991): 71-86. (Classic! just browse)

3. Martin CD, Porter MA. The extraordinary SVD. The American Mathematical Monthly. 2012 Dec 1;119(10):838-51.  (Excellent Review!)

Date: 01-16-2025

"If you don’t believe in yourself why is anyone else going to believe in you?" ― Tom Brady

• Eckhart-Young Theorem and Low Rank Approximations 

• Poincare Diagram: Stability diagram classifying Poincaré maps as stable or unstable according to their features

• Image Compression with Low-Rank SVD: Example

• My talk yesterday (1/15/2025)

Date: 01-21-2025

Quote of the Day

"Young man, in mathematics you don't understand things. You just get used to them" ― John von Neumann 

• Guest Lecture: Dr. Cleve Moler

MathWorks Co-Founder and Chief Mathematician Cleve B. Moler

• 1. How SVD Saved the Universe

  2. 1976 Matrix Singular Value Decomposition Film

  3. Direct link to YouTube:  https://www.youtube.com/watch?v=R9UoFyqJca8

Date: 1-23-2025

Quote of the Day

"Everything is practice" ― Pele 

• SVD and PCA

• Scree Plots and Optimal Hard Threshold (OHT)

• Low Rank Approximations 

• OHT and Clustering

• Mouse Data

Additional Reading

1. Gavish, Matan, and David L. Donoho. "The optimal hard threshold for singular values is 4/sqrt(3)." IEEE Transactions on Information Theory 60.8 (2014): 5040-5053.  (Amazing Paper!)

2. Excellent book chapter (Please read pages 31 - 32)

https://databookuw.com/

3. Donoho, David, Matan Gavish, and Elad Romanov. "ScreeNOT: Exact MSE-optimal singular value thresholding in correlated noise." The Annals of Statistics 51, no. 1 (2023): 122-148. 

4. Udell, Madeleine, and Alex Townsend. "Why are big data matrices approximately low rank?." SIAM Journal on Mathematics of Data Science 1.1 (2019): 144-160. 

5. Thibeault, Vincent, Antoine Allard, and Patrick Desrosiers. "The low-rank hypothesis of complex systems." Nature Physics 20, no. 2 (2024): 294-302. 

Date: 1-28-2025

Quote of the Day

"Nature has a great simplicity and therefore a great beauty" ― Richard Feynman 

• Poincare Diagram: Stability diagram classifying Poincaré maps as stable or unstable according to their features

• Inverse and Pseudoinverse of a Matrix

• Dynamic Mode Decomposition (DMD)

• DMD Demo Codes

Book

• Kutz, J. Nathan, et al. Dynamic mode decomposition: data-driven modeling of complex systems. Society for Industrial and Applied Mathematics, 2016. 

Chapter 1: Dynamic Mode Decomposition: An Introduction

Additional Reading

1. Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of fluid mechanics 656 (2010): 5-28. 

2. Schmid, Peter J. "Dynamic mode decomposition and its variants." Annual Review of Fluid Mechanics 54 (2022): 225-254. 

3. Tu, Jonathan H. "Dynamic mode decomposition: Theory and applications." PhD diss., Princeton University, 2013.

Date: 1-30-2025

Quote of the Day

"Be constantly on the lookout for hype"―David Heckerman

• Guest Lecture: Causal discovery from data (Dr. David Heckerman from Amazon)

• Please visit and like these GitHub:

  1. https://github.com/Jpickard1/BRAD

  2. https://github.com/Jpickard1/BRAD-Video

Date: 2-04-2025

Quote of the Day

"I think one of the things about creativity is not to be afraid of saying the wrong thing " ― Sydney Brenner 

• Data and Laplacian Matrix

• Edify Image

Papers

1. Von Luxburg, Ulrike. "A tutorial on spectral clustering." Statistics and computing 17.4 (2007): 395-416.  (Excellent Review!)

2. Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps and spectral techniques for embedding and clustering." Advances in neural information processing systems. 2002. 

3. Ng, Andrew Y., Michael I. Jordan, and Yair Weiss. "On spectral clustering: Analysis and an algorithm." Advances in neural information processing systems 2 (2002): 849-856.

4. Eliasof, Moshe, Eldad Haber, and Eran Treister. "Graph neural reaction diffusion models." SIAM Journal on Scientific Computing 46, no. 4 (2024): C399-C420. 

Date: 2-06-2025

Quote of the Day

"You have to spend some energy and effort to see the beauty of mathematics" ― Maryam Mirzakhani  

• Working with Data (PS2: 5 and 6)

  • Joshua Pickard, Cooper Stansbury, Amit Surana, Anthony Bloch, and Indika Rajapakse. "Biomarker Selection for Adaptive Systems." arXiv preprint arXiv:2405.09809 (2024). 

• Data Collections

• Graph Representation Learning

• Cooper Stansbury will present a Bioinformatics PhD Defense

Date: 2-11-2025

Quote of the Day

"Imagination will often carry us to worlds that never were. But without it we go nowhere" ― Carl Sagan

• Marc's DMD Slides

  • Joshua L. Proctor, Philip A. Eckhoff. "Discovering dynamic patterns from infectious disease data using dynamic mode decomposition." International Health, Volume 7, Issue 2, March 2015

• OHT revisited: Slides from Matan Gavish

• K-Means Clustering

• The leading eigenvector

Date: 2-13-2025

Quote of the Day

"I believe in my game, and I believe in me. At the end of the day, I'm my biggest fan" ― Serena Williams 

• Stories light the path to mastery: Professor Lee Hartwell

• Low rank matrices (Let's do it one more time!)

Papers

1. Udell, Madeleine, and Alex Townsend. "Why are big data matrices approximately low rank?." SIAM Journal on Mathematics of Data Science 1.1 (2019): 144-160.

Date: 02- 18-2025

"Stay hungry, stay foolish" ― Steve Jobs 

• Least-squares problems 

• Data-guided Control (DGC)

Papers

1. Ronquist S, Patterson G, Muir LA, Lindsly S, Chen H, Brown M, Wicha MS, Bloch A, Brockett R, Rajapakse I. "Algorithm for cellular reprogramming." Proceedings of the National Academy of Sciences. 2017 Nov 7;114(45):11832-7. Data-guided Control (DGC) Supporting Information and Slides from Scott Ronquist 

2. Proctor, Joshua L., Steven L. Brunton, and J. Nathan Kutz. "Dynamic mode decomposition with control." SIAM Journal on Applied Dynamical Systems 15, no. 1 (2016): 142-161. 

Date: 02- 20-2025

"I don’t think limits”  ― Usain Bolt

1. Spectral Clustering and Data-Guided Control (DGC)

2. Mathematics of DGC

3. Some additional materials

Date: 02- 25-2025

"The only way to learn mathematics is to do mathematics ” ― Paul Halmos 

Linear Dimension Reduction

• PCA and MDS

• Mathematics of DGC

• Some additional materials

Date: 02- 27-2025

"Simplicity makes me happy ” ― Alicia Keys 

Nonlinear Dimension Reduction

• Data Background for Motivation 

• Kullback–Leibler divergence 

  • Other Loss functions 

  • Good post that explains some common ones: https://www.geeksforgeeks.org/ml-common-loss-functions/ 

• Stochastic Neighbor Embedding and UMAP

Papers

1. UMPAP: McInnes, Leland, John Healy, and James Melville. "Umap: Uniform manifold approximation and projection for dimension reduction." arXiv preprint arXiv:1802.03426 (2018).

2. t-SNE: Van der Maaten, Laurens, and Geoffrey Hinton. "Visualizing data using t-SNE." Journal of machine learning research 9.11 (2008).

3. Wang, Y., Huang, H., Rudin, C. and Shaposhnik, Y., 2021. Understanding how dimension reduction tools work: an empirical approach to deciphering t-SNE, UMAP, TriMAP, and PaCMAP for data visualization. The Journal of Machine Learning Research, 22(1), pp.9129-9201. 

Remark: This is a really nice review in general, but the most relevant parts are pages 8, 9, and 10

Winter Vacation (March 1 - March 10, 2025): Suggested Reading!

1. Strogatz S. Sync: The emerging science of spontaneous order. 

2. uring, Alan Mathison. "The chemical basis of morphogenesis." Bulletin of mathematical biology 52.1-2 (1990): 153-197.