The scalable optimization of sensor placement is a central challenge for high-dimensional estimation and control. Nearly all downstream control decisions are affected by these sensor measurements. Exploiting low-dimensional structure inherent to the process physics is crucial for efficient optimization over complex environments. Our data-driven method for sensor placement minimizes error covariance using D-optimal design criteria, which provides an evaluation metric for a given sensor configuration and corresponding estimates of reconstruction uncertainty under noisy measurements. We show results for high-dimensional signal reconstruction in imaging, fluid flows, control, and manufacturing.