Can anyone please elucidate the difference between Euclidean and Non-Euclidean Space? I have seen some literatures where they haver mentioned social networks, sensor networks etc as examples of non-euclidean spaces. Is a power grid also an example of non-Euclidean?
Dear Mihir,
A non-Euclidean space is one where the basic assumptions about parallel lines do not hold. For instance, two parallel lines can intersect or move away from eachother in non-Euclidean spaces. Machine learning is ultimately interested in the manipulation of vectors within a space, so what do we do if the data that comprises these vectors is best understood in a non-Euclidean way? A classic example is a graph or tree. Let's say we want the volume (number of nodes) to grow exponentially with radius (number of layers). We need a hyperbolic non-Euclidean geometry to do this (like in the papers below).
https://arxiv.org/pdf/1705.08039.pdf
https://arxiv.org/pdf/1705.10359.pdf
Of particular interest these days are product manifolds that can create a composite space out of a cartesian product of base spaces. One researcher actually found that power grids can be understood much better in the context of a 5-hyperboloid and a 5-sphere (H5×S5) product manifold.
https://dawn.cs.stanford.edu/2019/10/10/noneuclidean/
Hope this helps!
Dear Mihir Malladi :
Euclidean space is the type of geometry with which we are most familiar.
Euclid, a Greek mathematician who lived around 300 BC, gave his name to Euclidean geometry. "The Elements," his work, is a compilation of axioms, theorems, and proofs about squares, circles, acute angles, isosceles triangles, and other topics. Euclid's 2000-year-old book contains the majority of the theorems taught in high schools today.
While any space that is not Euclidean is known as non-Euclidean space. Non-Euclidean geometry is a set of definitions, assumptions, and proofs that characterize objects such as points, lines, and planes. Spherical and hyperbolic geometry are the two most frequent non-Euclidean geometries.
- https://elikakurniadi.wordpress.com/2011/11/14/the-difference-between-euclidean-and-non-euclidean-geometry/
The power grid make considerably more sense when viewed from the perspective non-Euclidean space. Please have a look at these links:
- https://www.sciencedaily.com/releases/2015/05/150506084636.htm
- https://dawn.cs.stanford.edu/2019/10/10/noneuclidean/
- https://arxiv.org/pdf/1611.08097.pdf
Best Regards;
Ehtisham
Hello Nathan Zavanelli and Ehtisham Lodhi ,
I am confused. Your citations appear to be mostly concerned with animal or machine learning in non-Euclidean spaces. Which of these citations directly discusses electrical power systems, such as the national electrical power grid? And what problems, concerning electrical power grids, can only be solved by the non-Euclidean space approach that cannot be solved by the old 2-dimensional Euclidean representation of such a network?
Regards,
Thomas Cuff
Is a graph G=(N,E) N are the nodes and E the edges an example of a non-Euclidean geometrical structure?
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