Learning Transition Operators From Sparse Space-Time Samples

We consider the nonlinear inverse problem of learning a transition operator A from partial observations at
T
different times, in the form of sparse observations of entries of the powers A,A
²
,...,A
^T
. We address the nonlinearity of this spatio-temporal transition operator recovery problem by a suitable embedding into block-Hankel matrices, transforming it to a low-rank matrix completion problem, even when A has full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than
O
(
rn
log(
nT
)) space-time samples are sufficient to ensure accurate recovery of a rank-
r
operator A of size
n
×
n
. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order
O
(
rnT
) and periteration time complexity linear in
n
, and confirm in numerical experiments that for several graph transition operators, the theoretical findings accurately track empirical phase transitions.