NOTES, SLIDES, AND PAPERS

Date: 01-08-2026

• Introduction

Date: 01-15-2024

Quote of the Day

"The only way to do great work is to love what you do" ― Steve Jobs 

• The Extraordinary SVD: During the lecture, Dr. Cleve Moler will join the class virtually 

  1. How SVD Saved the Universe

  2. SVD Measures Partisanship in the U. S. Senate

  3. 1976 Matrix Singular Value Decomposition Film

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

Date: 01-20-2024

Quote of the Day

"One never notices what has been done; one can only see what remains to be done" ― Marie Curie 

• 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)

Date: 01-22-2024

Quote of the Day

"If you want to be the best, you have to do things that other people aren’t willing to do ― Michael Phelps 

Papers

1. Brin, Sergey, and Lawrence Page. "The anatomy of a large-scale hypertextual web search engine." (1998). 

2. 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. 

PatentsUS6285999B1

1. US6285999B1 Method for node ranking in a linked database. Lawrence Page: 1998-01-09

2. 115-008219-US-PS1 Network approach to navigating the human genome. Indika Rajapakse: submitted November 2020.

Date: 01-27-2026

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: This comes up all the time when you are dealing with real data 

• Dynamic Mode Decomposition (DMD) and 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: 01-29-2025

Quote of the Day

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

• Problem Set 1: Discussion and Grading 

• SVD, PCA and DMD again...

• Eckhart-Young Theorem

Papers

1. I would like to share this paper in case you haven’t seen it. Even if it’s not your area, it’s inspiring to see how AI models can achieve beautiful results. 

Avsec et. al. Advancing regulatory variant effect prediction with AlphaGenome. Nature 649, 1206–1218 (2026) 

2. Joshua Pickard, Cooper Stansbury, Amit Surana, Anthony Bloch, and Indika Rajapakse. "Biomarker Selection for Adaptive Systems." Proceedings of the National Academy of Sciences, (2025)

Date: 2-03-2026

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 Graphs

• Data and Laplacian Matrix

• Fiedler Number and Fiedler Vector

• Edify Image: Please check this website for applications of diffusion models 

Papers: I am sharing these references for completeness; you do not need to read them. I personally think it is helpful to know where these methods were originally introduced. 

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. Wiskott, Laurenz, and Fabian Schönfeld. "Laplacian matrix for dimensionality reduction and clustering." In European Big Data Management and Analytics Summer School, pp. 93-119. Cham: Springer International Publishing, 2019. 

Date: 02-05-2026

Quote of the Day

"Everything is practice" ― Pele 

• Problem Set 2 discussion

• A continuation of Tuesday ....

Date: 02-10-2026

Greg Harden ( – September 12, 2024) : Stay Sane in an Insane World: How to Control the Controllables and Thrive

I will review spectral clustering and DMD, then introduce DMD with control and preview data-driven ideas on controllability and observability. 

• Network Controllability: Slides Courtesy of Yang Liu

• Perturbation Theory and Condition Number

• Data-Guided Control

• Linear Control Theory (Book Chapter from: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control) 

Papers

1. Liu, Yang-Yu, Jean-Jacques Slotine, and Albert-László Barabási. "Controllability of complex networks." Nature 473.7346 (2011)

Date: 02-12-2026

Quote of the Day

"Enthusiasm is common. Endurance is rare " ― Angela Duckworth 

• PS2 and continuation of Tuesday ......

• The k− Means Algorithm and KL Divergence

• Centrality: Centrality helps identify key nodes in a network and understand their influence, aiding decisions and predictions 

• Synthetic Data and Data Collections (If you are aware of useful resources for time series data, please include them here: DATA 2026)

• Working Example

Date: 02-17-2026

Quote of the Day

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

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

• Heckerthoughts: Chapter 4 provides a step-by-step method for constructing a graphical model from data 

Date: 02-19-2026

Quote of the Day

"Order and simplification are the first steps toward the mastery of a subject" ― Thomas Mann 

• Mathematics of Optimization and CVX

• 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

Date: 02-24-2026

Quote of the Day

"I don't care that they stole my idea. I care that they don't have any of their own" ― Nikola Tesla 

• DGC: Data-Guided Control Algorithms and Potential Future Directions

• Network Controllability: Slides , Network Observability: Slides: Courtesy of Yang Liu: Please review these slide decks and familiarize yourself with the concepts of controllability and observability 

• A Taste of Koopman Theory

Papers

1. Egerstedt, Magnus. "Degrees of control." Nature 473, no. 7346 (2011): 158-159. 

2. Moore, Bruce. "Principal component analysis in linear systems: Controllability, observability, and model reduction." IEEE transactions on automatic control 26, no. 1 (2003): 17-32. 

Some classic papers for inspiration. Please browse them at your convenience....

1. Lorenz, Edward N. (1963). "Deterministic non-periodic flow". Journal of the Atmospheric Sciences. 20 (2): 130–141 

2. Koopman, B. O. (1931). "Hamiltonian Systems and Transformations in Hilbert Space". Proceedings of the National Academy of Sciences. 17 (5): 315–318 

Date: 02-26-2026

Quote of the Day

"It’s important to remember that AI is a tool, not a magic bullet" ― Demis Hassabis

• Foundation Models: GeneFormer and ARCH3D (Guest lecture by Dr. Nick Galioto)

• I was inspired by the Nobel Prize lecture by Demis Hassabis on “Accelerating Scientific Discovery with AI", and wanted to share it with you. Please take some time to browse through the slides!

Papers

1. C. V. Theodoris et al., “Transfer learning enables predictions in network biology,” Nature, vol. 618, no. 7965, pp. 616–624, Jun. 2023

More Papers to browse: TwinCell Blueprint

Date: 3-10-2026 and 3-17-2026 

Quote of the Day

"The true delight is in the finding out rather than in the knowing"―Isaac Asimov  

Tensors = Multi-Dimensional Arrays, Graphs represented with adjacency matrix and Hypergraphs represented with adjacency tensor 

• How to Decompose a Tensor: Examples 

• Some Introductory slides on Tensors and Hypergraphs 

• Tensors and Hypergraphs (Notes)

• Other Resources:  Dr. Charles Van Loan (Wonderful introduction to Tensors!)

• Multi-correlations and Hyperlink Prediction (Slides courtesy of Can Chen)

• Popular Hypergraphs: STEPHEN WOLFRAM  and Wolfram Physics Project Launch

Software

• MATLAB Tensor Toolbox

• pyTTB (Python Tensor Toolbox)

• HAT: Hypergraph Analysis Toolbox

Data Collection: Collection of raw, pairwise, and hypergraph data. Please let me know if you know of any additional resources. 

Papers

1. Kolda, Tamara G., and Brett W. Bader. "Tensor decompositions and applications." SIAM review 51.3 (2009): 455-500. (Excellent Review!)

2. Wolf, Michael M., Alicia M. Klinvex, and Daniel M. Dunlavy. "Advantages to modeling relational data using hypergraphs versus graphs." 2016 IEEE High Performance Extreme Computing Conference (HPEC). IEEE, 2016.

3. Pickard J,  Can C, Salman R, Stansbury C, Kim S, Surana A, Rajapakse I. “HAT: Hypergraph Analysis Toolbox,” PLOS Computational Biology, 2023 

4. Surana A, Chen C, Rajapakse I. "Hypergraph Similarity Measures."  IEEE Transactions on Network Science and Engineering,  2022 (Slides)

5. Donnat, Claire, and Susan Holmes. "Tracking network dynamics: A survey using graph distances." The Annals of Applied Statistics 12.2 (2018): 971-1012. 

Book

1. Ballard G, Kolda TG. Tensor Decompositions for Data Science. Cambridge University Press; 2025. 

Shape of Data: An Interesting Video (Think about why we need to go beyond graphs)

Missing Method: I believe this is very important!

Optimal Hard Threshold (OHT): A mathematically derived threshold for truncating singular values in matrix denoising. 

1. OHT: 1.  Slides from Matan Gavish 2. OHT and Clustering

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/

Date: 3-19-2026

Quote of the Day

"Information is the resolution of uncertainty”― Claude Shannon 

• Graph Entropy and Tensor Entropy: Entropy provides a quantitative measure of data’s inherent randomness and information content, informing feature engineering and model optimization!

• Eigenfaces and TensorFaces

• Eigenfaces and TensorFaces Implementation

Papers

1. Chen, Pin-Yu, et al. "Fast incremental von neumann graph entropy computation: Theory, algorithm, and applications." International Conference on Machine Learning. PMLR, 2019

2. Chen C, Rajapakse I. "Tensor Entropy  for Uniform Hypergraphs." IEEE Transactions on Network Science and Engineering 7.4 (2020): 2889-2900.

3. Tribus, Myron, and Edward C. McIrvine. "Energy and information." Scientific American 225, no. 3 (1971): 179-190. (Claude Shannon credited John von Neumann with providing valuable advice on naming his measure of information content)

4. MacArthur, Ben D., and Ihor R. Lemischka. "Statistical mechanics of pluripotency." Cell 154.3 (2013): 484-489. (An illustrative example of entropy in biological systems)

Date: 3-24-2026

Quote of the Day

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

Today, we will spend part of the class with a visitor (Dr. Ed Pagani) discussing the Ethics of data interpretation. Please also read the following section. 

• Please read pages 247 to 256 of Chapter 7,  Data-Driven Dynamical Systems (Brunton SL, Kutz JN. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press; 2019).  Please check out this beautiful video on SINDy! 

Papers

1. EDITORIAL, 19 February 2025: Why retractions data could be a powerful tool for cleaning up science

2. Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. "Discovering governing equations from data by sparse identification of nonlinear dynamical systems." Proceedings of the national academy of sciences 113.15 (2016): 3932-3937.

Date: 3-26-2026

Quote of the Day

"I never, never in my life took a course in economics " ― Lloyd Shapley ( 2012 Nobel Prize for Economics) 

First Part of Class: Guest Lecture by Kathryn Moler 

Genesis Mission

Second Part of Class: SINDy and Final Projects

Papers

1. EDITORIAL, 25 Nov 2025: Gil, Darío, and Kathryn A. Moler. "Accelerating science with AI." Science 390, no. 6777 (2025): 965-965. 

2. Brunton, Steven L., Joshua L. Proctor, and J. Nathan Kutz. "Discovering governing equations from data by sparse identification of nonlinear dynamical systems." Proceedings of the national academy of sciences 113.15 (2016): 3932-3937.

Date: 3-31-2026

Quote of the Day

"The Cube is, at the same time, a symbol of simplicity and complexity" ― Erno Rubik 

Today, I will discuss the mathematics of nonlinear dimensionality reduction and how we can visualize data to uncover patterns. We will also have an in-person visitor from NVIDIA, Dr. Angus Forbes, who will present advanced data visualization methods. 

Dimension Reduction of Data (PCA, t-SNE, UMAP)

Data Visualization Resources

Date: 04-02-2026

Quote of the Day

"The more we know, the more we realize there is to know" ― Jennifer Doudna

• Introduction: These slides deck includes slides from my good friend Yuan Yao

• A Taste of Topology

• Chapter 2: Charles Pugh. Real mathematical analysis

Papers

1. Wasserman L. Topological data analysis. Annual Review of Statistics and Its Application. 2018 Mar 7;5:501-32. 

2. R. Ghrist, “Barcodes: The persistent topology of data,” Bull. Amer. Math. Soc., vol. 45, no. 1, pp. 61–75, 2008, doi: 10.1090/S0273-0979-07-01191-3.

3. Lum PY, Singh G, Lehman A, Ishkanov T, Vejdemo-Johansson M, Alagappan M, Carlsson J, Carlsson G. Extracting insights from the shape of complex data using topology. Scientific reports. 2013 Feb 7;3(1):1-8. 

4. Derenick, Jason, Alberto Speranzon, and Robert Ghrist. "Homological sensing for mobile robot localization." In 2013 IEEE International Conference on Robotics and Automation, pp. 572-579. IEEE, 2013.