  Learning hydrologic models for accurate riverine flood prediction at scale is
a challenge of great importance. One of the key difficulties is the need to
rely on in-situ river discharge measurements, which can be quite scarce and
unreliable, particularly in regions where floods cause the most damage every
year. Accordingly, in this work we tackle the problem of river discharge
estimation at different river locations. A core characteristic of the data at
hand (e.g. satellite measurements) is that we have few measurements for many
locations, all sharing the same physics that underlie the water discharge. We
capture this scenario in a simple but powerful common mechanism regression
(CMR) model with a local component as well as a shared one which captures the
global discharge mechanism. The resulting learning objective is non-convex, but
we show that we can find its global optimum by leveraging the power of joining
local measurements across sites. In particular, using a spectral initialization
with provable near-optimal accuracy, we can find the optimum using standard
descent methods. We demonstrate the efficacy of our approach for the problem of
discharge estimation using simulations.
