  We introduce an optimization model for maximum likelihood-type estimation
(M-estimation) that generalizes a large class of existing statistical models,
including Huber's concomitant M-estimator, Owen's Huber/Berhu concomitant
estimator, the scaled lasso, support vector machine regression, and penalized
estimation with structured sparsity. The model, termed perspective
M-estimation, leverages the observation that convex M-estimators with
concomitant scale as well as various regularizers are instances of perspective
functions. Such functions are amenable to proximal analysis, which leads to
principled and provably convergent optimization algorithms via proximal
splitting. Using a geometrical approach based on duality, we derive novel
proximity operators for several perspective functions of interest. Numerical
experiments on synthetic and real-world data illustrate the broad applicability
of the proposed framework.
