The Optimal Smoothings of Sublinear Functions and Convex Cones
Published in ArXiv Preprint, 2025
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For any sublinear function f (e.g the ReLU function, any norm, the max function, and the maximum eigenvalue function), the paper provides formulae for smooth, convex functions that minimize distance to f among smooth, convex functions of bounded curvature. Similarly, formulae for convex sets that minimize distance to any given convex cone are provided. These smooth approximations have a wide range of applications, particularly in optimization, where they can be used to solve optimization problems faster.
Recommended citation: Thabo Samakhoana, Benjamin Grimmer. (2025). "The Optimal Smoothings of Sublinear Functions and Convex Cones." ArXiv .
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