  Sensor placement and feature selection are critical steps in engineering,
modeling, and data science that share a common mathematical theme: the selected
measurements should enable solution of an inverse problem. Most real-world
systems of interest are nonlinear, yet the majority of available techniques for
feature selection and sensor placement rely on assumptions of linearity or
simple statistical models. We show that when these assumptions are violated,
standard techniques can lead to costly over-sensing without guaranteeing that
the desired information can be recovered from the measurements. In order to
remedy these problems, we introduce a novel data-driven approach for sensor
placement and feature selection for a general type of nonlinear inverse problem
based on the information contained in secant vectors between data points. Using
the secant-based approach, we develop three efficient greedy algorithms that
each provide different types of robust, near-minimal reconstruction guarantees.
We demonstrate them on two problems where linear techniques consistently fail:
sensor placement to reconstruct a fluid flow formed by a complicated
shock-mixing layer interaction and selecting fundamental manifold learning
coordinates on a torus.
