  Optimal sensor placement is a central challenge in the design, prediction,
estimation, and control of high-dimensional systems. High-dimensional states
can often leverage a latent low-dimensional representation, and this inherent
compressibility enables sparse sensing. This article explores optimized sensor
placement for signal reconstruction based on a tailored library of features
extracted from training data. Sparse point sensors are discovered using the
singular value decomposition and QR pivoting, which are two ubiquitous matrix
computations that underpin modern linear dimensionality reduction. Sparse
sensing in a tailored basis is contrasted with compressed sensing, a universal
signal recovery method in which an unknown signal is reconstructed via a sparse
representation in a universal basis. Although compressed sensing can recover a
wider class of signals, we demonstrate the benefits of exploiting known
patterns in data with optimized sensing. In particular, drastic reductions in
the required number of sensors and improved reconstruction are observed in
examples ranging from facial images to fluid vorticity fields. Principled
sensor placement may be critically enabling when sensors are costly and
provides faster state estimation for low-latency, high-bandwidth control.
MATLAB code is provided for all examples.
