From Discrete Measurements to Bounded Gradient Estimates: A Look at Some Regularizing Structures

Obtaining a reliable gradient estimate for an unknown function when given only its discrete measurements is a common problem in many engineering disciplines. While there are many approaches to obtaining an estimate of a gradient, obtaining lower and upper bounds on this estimate is an issue that is often overlooked, as rigorous bounds that are not overly conservative usually require additional assumptions on the function that may either be too restrictive or impossible to verify. In this work, we try to make some progress in this direction by considering four general structural assumptions as a means of bounding the function gradient in a rigorous likelihood sense. After proposing an algorithm for computing these bounds, we compare their accuracy and precision across different scenarios in an extensive numerical study.