  Much attention has been devoted to understanding the microscopic pathways of
phase transition between two equilibrium condensed phases (such as liquids and
solids). However, the microscopic pathways between non-equilibrium,
non-diffusive amorphous (glassy) phases still remain poorly understood. In this
work, we have employed computer simulations, persistence homology (a tool
rooted in topological data analysis), and machine learning to probe the
microscopic pathway of pressure-induced non-equilibrium transition between the
low- and high-density amorphous (LDA and HDA, respectively) ice phases of
TIP4P/2005 and ST2 water models. Using persistence homology and machine
learning, we introduced a new order parameter that unambiguously identifies the
LDA and HDA-like local environments. The system transitions continuously and
collectively in the order parameter space via a pre-ordered intermediate phase
during the compression of the LDA phase. The local order parameter
susceptibilities show a maximum near the transition pressure ($P^*$) --
suggesting maximum structural heterogeneities near $P^*$. The HDA-like clusters
are structurally ramified and spatially delocalized inside the LDA phase near
the transition pressure. We have further investigated the (geometrical)
structures and topologies of the LDA and HDA ices formed via different
protocols and also studied the dependence of the microscopic pathway of phase
transition in the order parameter space on the protocol followed to prepare the
initial LDA phase. Finally, the method adopted here to study the microscopic
pathways of transition is not restricted to the system under consideration and
provides a robust way of probing phase transition pathways involving any two
condensed phases at both equilibrium and out-of-equilibrium conditions.
