In high-dimensional statistics, finding the maximum likelihood estimate associated with a statistical model is often associated with solving a (convex) non-smooth optimization problem. One particular model for maximum likelihood-type estimation (M-estimation) which generalizes a large class of well-known estimators, including Huber's concomitant M-estimators and the scaled Lasso, is the perspective M-estimation model. Perspective M-estimation leverages the observation that convex M-estimators with concomitant scale as well as various regularizers are instances of perspective functions and is thus amenable to efficient global optimization. We extend this model to allow for regression models with compositional covariate data which are commonplace in biology, including microbiome and metabolomics data. We introduce new perspective M-estimators that can handle outliers in outcome variables and heteroscedasticity in the covariates and show how to solve the associated non-smooth optimization problem with proximal algorithms. We find excellent empirical performance of the estimators on synthetic and real-world prediction tasks involving human gut and marine microbiome data.