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A sheaf is a combinatorial, high-dimensional data structure on a topological space. You can use it to stitch together images, spatio-temporal target tracking, i
by espeed 10y ago
A sheaf is a combinatorial, high-dimensional data structure on a topological space. You can use it to stitch together images, spatio-temporal target tracking, inference, data fusion, IoT time-series, continuous functions -- all kinds of things. Crazy cool stuff!
Applied sheaf theory is pretty new, and not yet widely known. This DARPA sheaf tutorial is a full two-day video series by Prof Michael Robinson (great teacher), and it's about the equivalent of a semester CS course compressed into two days. If you want to see an intro/big-picture overview video on what the sheaf data structure is, and how it's related to graph databases, topology, category theory, and data analysis, see:
"Sheaves for engineering problems" https://www.youtube.com/watch?v=223-0x2KNOg https://www.youtube.com/watch?v=223-0x2KNOg
And for a deep dive into a paper, see:
"Sheaves are the canonical data structure for sensor integration" https://arxiv.org/abs/1603.01446 https://arxiv.org/abs/1603.01446
Python Cellular Sheaf Library:
https://github.com/kb1dds/pysheaf https://github.com/kb1dds/pysheaf
- defen 10y agoThat youtube link is great, thanks.
- faizshah 10y agoHave you found a book or a review/survey paper on this topic? I just started Computational Homology by Kaczynski et al. but I just looked in the index and didn't find anything on sheaves.
- espeed 10y agoProf Robinson's book is Topological Signal Processing [1], and there are a ton of papers linked from his home page [2]. For a general overview, check out the wikipedia page: https://en.wikipedia.org/wiki/Sheaf_(mathematics) https://en.wikipedia.org/wiki/Sheaf_(mathematics) [1] https://www.amazon.com/Topological-Signal-Processing-Mathematical-Engineering/dp/364236103X https://www.amazon.com/Topological-Signal-Processing-Mathema... [2] http://www.drmichaelrobinson.net/publications.html http://www.drmichaelrobinson.net/publications.html
- sn9 10y agoRobert Ghrist offers his Elementary Applied Topology for free on his site. It's useful more as a supplement or survey than a primary text to learn. https://www.math.upenn.edu/~ghrist/notes.html https://www.math.upenn.edu/~ghrist/notes.html
- diamonis 10y agohttp://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html http://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html
- fmap 10y agoSheaf models are one of the most useful tools in logic and I'll definitely read this paper once I have more time again. One thing that seems strange to me based on the overview video is the focus on sheafs on a topological space. In mathematics it's typically easier and more flexible to consider sheafs on categories equipped with a Grothendieck coverage. Is there a reason why topological spaces are sufficient to model sensor integration?
- auggierose 10y agoMaybe because topological spaces are more familiar to many mathematicians than categories?
- hackernewsacct 10y agoWhat kind of math background do you need to understand Sheaf theory? Are these lectures truly accessible for someone with a non-background? I tried watching bits of the youtube video OP linked but never saw where he actually defined what a Sheaf was as oppose to building motivation of learning it for applications. I am excited for any truly accessibly beginner resources. Want to get your opinion if you links are for the true beginner?