Abstract
The relationships between the structural topology of artificial neural networks, their computational flow, and their performance is not well understood. Consequently, a unifying mathematical framework that describes computational performance in terms of their underlying structure does not exist. This paper makes a modest contribution to understanding the structure-computational flow relationship in artificial neural networks from the perspective of the dicliques that cover the structure of an artificial neural network and the Forman-Ricci curvature of an artificial neural network’s connections. Special diclique cover digraph representations of artificial neural networks useful for network analysis are introduced and it is shown that such covers generate semigroups that provide algebraic representations of neural network connectivity.
Highlights
In recent years there has been an exponential growth in research focused upon machine learning and its application to those classes of real world problems that cannot be addressed using traditional computer algorithms
Graph theory and algebraic topology have been used with some success to analyze local and global network properties, e.g. [6,7], a unifying mathematical framework that describes the computational performance of an artificial neural network (ANN) in terms of its underlying structure does not yet exist
This paper makes a modest contribution to understanding the structure-computational flow relationship in ANNs from the perspective of the dicliques, e.g. [8], that cover the structure of an ANN and the FormanRicci (FR) curvature, e.g. [9,10], of an ANN’s connections
Summary
In recent years there has been an exponential growth in research focused upon machine learning and its application to those classes of real world problems that cannot be addressed using traditional computer algorithms. FR curvature provides a measure of the total amount of “computational flow” – or more flow - through a network node and quantifies the divergence of the flow emerging from a connection – i.e., the more negative the curvature, the more the divergence of the emergent flow, and, intuitively, the more wide spread the influence the connection has on the network Both curvature and flow – along with the underlying connection topology – provide a characterization of the relationship between computation and the structure of an ANN. Judicious assignments of FR curvatures and flows to the cover’s dicliques and wiring-diagram connections – or to the diclique digraph’s vertices and arcs provide high level insights into the general importance of subnetwork and node contributions to an ANN computation. Not addressed here, time series of such diclique representations obtained as computations propagate through the layers of an ANN can provide additional insights into the relative importance of maximal subnetworks and nodes to ANN computation
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