A Waddington landscape for prototype learning in generalized Hopfield
  networks
Nacer Eddine Boukacem Allen Leary Robin Thériault Felix Gottlieb Madhav Mani Paul François
Networks in machine learning offer examples of complex high-dimensional
dynamical systems reminiscent of biological systems. Here, we study the
learning dynamics of Generalized Hopfield networks, which permit a
visualization of internal memories. These networks have been shown to proceed
through a 'feature-to-prototype' transition, as the strength of network
nonlinearity is increased, wherein the learned, or terminal, states of internal
memories transition from mixed to pure states. Focusing on the prototype
learning dynamics of the internal memories we observe a strong resemblance to
the canalized, or low-dimensional, dynamics of cells as they differentiate
within a Waddingtonian landscape. Dynamically, we demonstrate that learning in
a Generalized Hopfield Network proceeds through sequential 'splits' in memory
space. Furthermore, order of splitting is interpretable and reproducible. The
dynamics between the splits are canalized in the Waddington sense -- robust to
variations in detailed aspects of the system. In attempting to make the analogy
a rigorous equivalence, we study smaller subsystems that exhibit similar
properties to the full system. We combine analytical calculations with
numerical simulations to study the dynamical emergence of the
feature-to-prototype transition, and the behaviour of splits in the landscape,
saddles points, visited during learning. We exhibit regimes where saddles
appear and disappear through saddle-node bifurcations, qualitatively changing
the distribution of learned memories as the strength of the nonlinearity is
varied -- allowing us to systematically investigate the mechanisms that
underlie the emergence of Waddingtonian dynamics. Memories can thus
differentiate in a predictive and controlled way, revealing new bridges between
experimental biology, dynamical systems theory, and machine learning.