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We introduce a class of quadratic support (QS) functions, many of which playa crucial role in a variety of applications, including machine learning, robuststatistical inference, sparsity promotion, and Kalman smoothing. Well knownexamples include the l2, Huber, l1 and Vapnik losses. We build on a dualrepresentation for QS functions using convex analysis, revealing the structurenecessary for a QS function to be interpreted as the negative log of aprobability density, and providing the foundation for statisticalinterpretation and analysis of QS loss functions. For a subclass of QSfunctions called piecewise linear quadratic (PLQ) penalties, we also developefficient numerical estimation schemes. These components form a flexiblestatistical modeling framework for a variety of learning applications, togetherwith a toolbox of efficient numerical methods for inference. In particular, forPLQ densities, interior point (IP) methods can be used. IP methods solvenonsmooth optimization problems by working directly with smooth systems ofequations characterizing their optimality. The efficiency of the IP approachdepends on the structure of particular applications. We consider the class ofdynamic inverse problems using Kalman smoothing, where the aim is toreconstruct the state of a dynamical system with known process and measurementmodels starting from noisy output samples. In the classical case, Gaussianerrors are assumed in the process and measurement models. The extendedframework allows arbitrary PLQ densities to be used, and the proposed IPapproach solves the generalized Kalman smoothing problem while maintaining thelinear complexity in the size of the time series, just as in the Gaussian case.This extends the computational efficiency of classic algorithms to a muchbroader nonsmooth setting, and includes many recently proposed robust andsparse smoothers as special cases.

Sparse/Robust Estimation and Kalman Smoothing with Nonsmooth Log-Concave Densities: Modeling, Computation, and Theory