Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties

Quantum Finite Automata and Quiver Algebras

“…In [5], we formulated noncommutative differential forms on a near-ring Ã, which induce Map(F, F)-valued differential forms on the moduli [R/GL(V)]. It is extended from the Karoubi-de Rham complex [6][7][8][9] for algebras to near-rings.…”

“…The original formulation of deep learning is over the flat vector space of representations Hom alg (CQ, End(V)), rather than the moduli space M = R // G of framed representations which has a semi-positive metric H T . In [5], we found the following way of connecting our new approach with the original approach by varying the bundle metric H i in (4).…”

“…Ignoring the last item (5) for the moment, this coincides with Definition 1 by setting G = Σ , the free group generated by a set Σ, and fixing an initial vector q 0 ∈ H V .…”

“…Remark 1. In the recent past there has been rising interest in the relations between machine learning and quiver representations [5,[18][19][20]. Here, we simply put quiver representation as a part of the formulation of an artificial neural network.…”

“…. , −1), we obtain a semi-negative moduli space (M − , H T ) [5], which is a generalization of the hyperbolic spaces or non-compact dual of Grassmannians. This is a very useful setting as there have been several fascinating works done on machine learning performed over hyperbolic spaces, for example [24][25][26][27].…”