c-lasso -- a Python package for constrained sparse and robust regression and classification

Abstract

We introduce c-lasso, a Python package that enables sparse and robust linear regression and classification with linear equality constraints. The underlying statistical forward model is assumed to be of the following form: \[ y = X β+ σε\qquad \textrm{subject to} \qquad Cβ=0 \] Here, $X\in {\mathbb{R}}^{n\times d}$is a given design matrix and the vector $y\in {\mathbb{R}}^n$ is a continuous or binary response vector. The matrix $C$ is a general constraint matrix. The vector $\beta \in {\mathbb{R}}^d$ contains the unknown coefficients and $\sigma$ an unknown scale. Prominent use cases are (sparse) log-contrast regression with compositional data $X$, requiring the constraint $1_d^T\beta =0$ (Aitchion and Bacon-Shone 1984) and the Generalized Lasso which is a special case of the described problem (see, e.g, (James, Paulson, and Rusmevichientong 2020), Example 3). The c-lasso package provides estimators for inferring unknown coefficients and scale (i.e., perspective M-estimators (Combettes and Müller 2020a)) of the form \[ \min_{β\in \mathbb{R}^d, σ\in \mathbb{R}_{0}} f\left(Xβ- y,σ \right) + λ\left\lVert β\right\rVert_1 \qquad \textrm{subject to} \qquad Cβ= 0 \] for several convex loss functions $f(\cdot ,\cdot )$. This includes the constrained Lasso, the constrained scaled Lasso, and sparse Huber M-estimators with linear equality constraints.