As I continue to explore applications of machine learning in complex systems modeling, I find myself sniffing below at the roots - searching for the geometric forms that underly complex system dynamics. I've embarked on the long journey of requisite coursework to equip me to leverage differential geometry in mapping complex networks, but am already beginning to stir up some intuition. For instance, it is has become clear to me, at least conceptually, that the geometry describing the dynamics of a system exist independently of whether or not we can deduce such geometry from the data we have at hand. In this light, the entire process seems to take on the feeling of archeology. A gentle dusting away of uncertainty to reveal a buried topology.
The following quote from John Holland's Signals and Boundaries comes to mind: "The first step toward a signal/boundary theory, then, is to phrase the questions we would like to address in ways that suggest premises for a deductive system". In my case, I'm curious about using the insights gleaned from our specific applications of machine learning predictors/classifiers to fill in more details of the underlying dynamic that underscores the phenomena we are observing. Can we work from the inside out in such cases, going beyond mere estimators for specific surface vectors? A theory to explain everything? No, a geometry. Albeit one intimately connected to countless other geometries, as demonstrated by the work of Maria Serrano and colleagues.