Discrete Differential Geometry: An Applied Introduction [pdf]
cs.cmu.edu
cs.cmu.edu
From page 8: "The manifold assumption is powerful because it lets us translate many of the things we know how to do in flat Euclidean spaces (e.g., work with vectors, differentiate, integrate, etc.) to more interesting curved spaces."
To be clear, it's not that I have worked with this in the past and merely forgot or something like that. I was thinking about linear transformations and essentially had the idea of taking that power to curved spaces and had no clue what would get me there or if that was even possible. Thank you for posting this. You saved me a lot of trouble.
Here's another example - https://hackaday.com/2018/12/22/clever-wedges-that-will-incr...
Try to find this article without filtering for hackaday domain and not using the exact title. tl;dr of article - these are triangle / wedge shaped stencil holes for SMD surface mount soldering on square / rectangular pads.
Collection of personal information that can be monetized, and ad clicking.
Everything about the Google search is easily understandable in terms of these fundamentals.
https://www.cs.cmu.edu/~kmcrane/
There's also a repo for this
This sort of thing intrigues me tremendously and I'm going over the text. At the same time, I always have a certain "what could you really get" feeling about systems that begin with machinery from objects with lots of structure (differentiable manifolds) and generalize and generalize until it is dealing a structure that seems utterly arbitrary. I mean, locally, "almost everywhere" (and similar caveats), the characteristics of a point of a differentiable manifold determine "nearly everything" about the points in its neighborhood. Oppositely, one node of graph no necessary relation to the next node.
So what exactly do we get from our complex machinery? Are the theorem ultimately more about "summation processes on graphs" than graphs?
http://www.reproducibility.org/RSF/book/bei/conj/paper_html/...
http://groups.csail.mit.edu/gdpgroup/6838_spring_2019.html
Roughly about measuring properties like curvature and distance on shapes, and similarity between shapes. Very cool class.
I highly recommend the hour-long lecture linked there.
What does the discrete version get you? What are its applications to CS?
i never explicitly ran into anything about discrete differential geometry, but that doesn't mean it wasn't there all along, lurking beneath (or perhaps above?) my level of understanding.
some reading: Evans -- Partial Differential Equations, Wendland -- Scattered Data Approximation, Wahba -- Spline models for observational data .
[0] "Anisotropic Polygonal Remeshing"
Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Lévy, Mathieu Desbrun
Everything! When you are programming a computer, everything must be discrete. If you need any differential geometry, it is discrete differential geometry then. You may want to hide this fact and pretend that your stuff is continuous, but at some point you will be computing derivatives by evaluating a function on nearby points. In that case, discrete differential geometry tells you which weights to put in your difference scheme.
This is a bit oversimplified/exaggerated.
We need to use discrete bits in our representation for a computer, but our numbers can be the coefficients of continuous functions or relations (e.g. polynomials or trigonometric polynomials), and so it is possible to represent continuous functions to whatever precision we have compute resources to handle without “discretizing” per se.
A simplicial complex is a bunch of simplices stuck together (adjacent simplices might share a lower-dimensional simplex as a common boundary).
https://en.wikipedia.org/wiki/Simplicial_complex
I’m not convinced “abstract” is a good word here, but what they mean is that it doesn’t have any metrical/geometric relationships beyond the graph structure.
package> https://github.com/chakravala/Grassmann.jl YouTube> https://www.youtube.com/watch?v=eQjDN0JQ6-s PDF doc> https://www.dropbox.com/sh/tphh6anw0qwija4/AAACiaXig5djrLVAK...
in what sense? i love when people say pretentious things like this to sound authoritative.
just because you take derivatives doesn't mean it's calculus. DG is not calculus - DG is calculus in spaces aren't globally flat but are locally flat. very different.
>The bar is just too high for normal people
in most schools that have DG classes they're junior level. certainly this class is a junior level class
http://brickisland.net/DDGSpring2019/grading-policy/
(no i didn't attend CMU or any similar tier school so i'm not speaking from a place of dunning-Kruger)
>It would indeed be interesting to see a coherent description of ML in a DG framework.
there is no need for such a thing - you don't need the machinery of connections, bundles, christoffel symbols, whatever else in order to take derivatives. you use those tools to be able to take derivatives in places where you can't do freshman calculus, not the other way around (bring those tools to places where you do do freshman calculus). it makes no sense.
it's like reasoning that because wiles used algebraic geometry to resolve fermat's last theorem that solving quadratic equations is really about algebraic geometry.
There are quite a few places differential geometry is very useful to know. Generally, if you want to know at a fundamental level "what does it mean to learn, what is inference truly?". You will find yourself in dire need of learning differential geometry. The easiest example is: the deeper your understanding of differential geometry, the more you'll be able to reason about hamiltonian monte carlo algorithms.
Information geometry also applies differential geometry, where you can think of learning as trajectories on a statistical manifold.
K-FAC, mirror descent and the natural gradient also derive from or are closely connected to work in information geometry. There's recent work connecting optimal transport. Optimal transport is an important idea and pops up in many surprising places, from GANs to programming language theory via way of modeling concurrency, for example (Kantorovich metric for bisimulation). Understanding differential geometry allows you to see and navigate such rich connections at a deep level. I heartily recommend it. A good place to start is: https://metacademy.org/roadmaps/rgrosse/dgml
https://arxiv.org/pdf/1410.3831v1.pdf
that's occupies very rarefied air, but that's not what the person to whom i'm responding to claims (that somewhere someone there's a relationship):
>DG is the foundation of all modern AI/ML/DL
which is just blather spoken in a tone of confidence in order to sound very smart.