Up until now, we’ve focused mainly on low-level concepts and definitions—for our next reading, we’ll take a high-level look at some very cool, contemporary algorithmic tools that leverage simplicial topology to make sense of large data sets in high dimensions. These tools fall under the broad heading of topological data analysis (TDA). One particular tool that’s received a lot of buzz in recent years (both within computer science as well as in applications in material science, imaging, biology, etc.) is persistent homology. The following video provides an excellent introduction, and should be mostly understandable based on what you’ve already learned in class:
Your reading assignment between now and February 2nd is to find one paper or survey on persistent homology that piques your interest (could be theory, could be applications) and summarize it to whatever degree you feel capable. Don’t worry if a lot of the terminology still seems alien—an important skill in reading academic papers is being able to extract the key idea without decoding every little detail and definition. If you find this subject interesting, you might consider doing your course project on TDA. More broadly, the emerging field of computational topology spans a broad range of beautiful and fascinating topics that could make good course projects.
Here are some pointers to get you started with persistent homology:
- Persistent Homology – A Survey – broad overview from a couple of its progenitors.
- Barcodes: The Persistent Topology of Data – another survey, with some nice visualizations (and featuring the Klein bottle)
- Three Examples of Applied & Computational Homology – nice short writeup of a few key applications in sensing and data analysis
- Computing Persistent Homology – one of the seminal papers on algorithms for persistent homology; will be a tough read unless you already have some experience in this stuff.
- Any number of applications in computational biology, natural language processing, computer vision… basically if you Google for “persistent homology” plus “your favorite subject,” you will likely turn up something interesting.
You might also try to give a rough summary of the different types of complexes that show up in topological data analysis (e.g., Vietoris-Rips, Čech, Witness, …).
Submission: As usual, please send an email to firstname.lastname@example.org and email@example.com no later than 10:00 AM on Tuesday, February 2nd including the string DDGSpring2016 in your subject line. Your email for readings should always include:
- a short (2-3 sentence) summary of what you read, and
- at least one question about something you found confusing / interesting / incomplete / not addressed.