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I'm interested in the industrial use of GDL (see https://arxiv.org/abs/1611.08097). Is it used in industry? That is, does any company have access to non-Euclidean data and process it directly instead of converting it to a more standard format?

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    $\begingroup$ GDL is a relatively new field of study. Apparently, the first graph NN model was introduced in 2005 by Scarselli et al. and only recently GDL models have started to be promising, so I don't think there are a lot of industries that have already adopted it. Maybe the chemical industry. $\endgroup$ – nbro May 6 at 21:11
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Recursive artificial networks are a type aligned well with GDL, and it is an increasingly popular one. Using non-orthogonal geometries is natural when using directed graphs as semantic association, data flow, or composition networks.

This more free-form approach enters into industry slowly because most programming languages use orthogonal structures designed around multidimensional arrays and nested loops to iterate through them, made popular in the FORTRAN era. It is common that a student learns to loop through an array before learning to call a function or test and branch. Computer science is, in some ways, entrenched in the orthogonal structure.

To call the two categories Euclidean and non-euclidean is inaccurate. The initial trend in machine learning followed a Cartesian paradigm, one locked into 90° angles in data structures. The free-form graph is non-cartesian but still just as Euclidean, as Euclid worked with angles other than 90° frequently.

Apple and Google are likely candidates for this seepage of recursive networks into industry data centers, for NLP applications. Language associations are free-form even though these structures are serialized for speech and written text.

Cognition is also non-cartesian. Graph based optimization (pre-GDL) has been part of cognitive research since the 1970s, especially in the LISP space at MIT and CMU and the U.S. Naval Research facilities for strategic analysis, Optimal strategies for a class of constrained sequential problems by JB Kadane, HA Simon, The Annals of Statistics, 1977. Algorithm 2 is a convergence strategy, but without layers of cells, yet the searching concepts of gradient are already present in the maximal and minimal operations in the search in steps c and f.

The aeronautics and auto industry is a likely candidates for seepage into the embedded computing for automated driving and flight. Most of this is either company confidential or classified.

This answer to the question about topological sophistication provides some background why non-euclidean space may be a more natural way to map associations, flow, and composition.

These are some practical applications.

We've used recursive networks to align complex models with data acquired during laboratory experimentation and hope to publish on that approach as part of another publication. It is effective, since the relationships between features in the real world form squares and cubes only by chance, so it is not very common. Research into materials and energy cannot be confined by Cartesian conceptions, even if the graphs published are on Cartesian coordinate axes, so that others with traditional analytic geometry backgrounds can understand the graphs quickly.

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