  Large-dimensional empirical data in science and engineering frequently has
low-rank structure and can be represented as a combination of just a few
eigenmodes. Because of this structure, we can use just a few spatially
localized sensor measurements to reconstruct the full state of a complex
system. The quality of this reconstruction, especially in the presence of
sensor noise, depends significantly on the spatial configuration of the
sensors. Multiple algorithms based on gappy interpolation and QR factorization
have been proposed to optimize sensor placement. Here, instead of an algorithm
that outputs a singular "optimal" sensor configuration, we take a thermodynamic
view to compute the full landscape of sensor interactions induced by the
training data. The landscape takes the form of the Ising model in statistical
physics, and accounts for both the data variance captured at each sensor
location and the crosstalk between sensors. Mapping out these data-induced
sensor interactions allows combining them with external selection criteria and
anticipating sensor replacement impacts.
