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: \[ y = X \beta + \sigma \epsilon \qquad \textrm{subject to} \qquad C\beta=0 \] 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 M\"uller 2020a)) of the form \[ \min_{\beta \in \mathbb{R}^d, \sigma \in \mathbb{R}_{0}} f\left(X\beta - y,{\sigma} \right) + \lambda \left\lVert \beta\right\rVert_1 \qquad \textrm{subject to} \qquad C\beta = 0 \] 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.

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c-lasso——一个用于约束稀疏和鲁棒回归和分类的 Python 包

我们引入了 c-lasso，这是一个 Python 包，它支持具有线性等式约束的稀疏和健壮的线性回归和分类。假设底层统计正向模型具有以下形式： \[ y = X \beta + \sigma \epsilon \qquad \textrm{subject to} \qquad C\beta=0 \] 这里， $X \in \ mathbb{R}^{n\times d}$ 是给定的设计矩阵，向量 $y \in \mathbb{R}^{n}$ 是连续或二元响应向量。矩阵 $C$ 是一个通用约束矩阵。向量 $\beta \in \mathbb{R}^{d}$ 包含未知系数，$\sigma$ 包含未知比例。突出的用例是（稀疏）对数对比回归与成分数据 $X$，需要约束 $1_d^T \beta = 0$（Aitchion 和 Bacon-Shone 1984）和广义套索，这是所描述的一个特例问题（见，例如，（詹姆斯，保尔森，和 Rusmevichientong 2020)，示例 3)。c-lasso 包提供了用于推断未知系数和尺度的估计器（即透视 M 估计器（Combettes 和 M\"uller 2020a）），形式为 \[ \min_{\beta \in \mathbb{R}^d, \sigma \in \mathbb{R}_{0}} f\left(X\beta - y,{\sigma} \right) + \lambda \left\lVert \beta\right\rVert_1 \qquad \textrm{subject to} \qquad C\beta = 0 \] 用于几个凸损失函数 $f(\cdot,\cdot)$。这包括受约束的套索、受约束的缩放套索和具有线性等式约束的稀疏 Huber M 估计器。{\sigma} \right) + \lambda \left\lVert \beta\right\rVert_1 \qquad \textrm{subject to} \qquad C\beta = 0 \] 几个凸损失函数 $f(\cdot,\cdot )$。这包括受约束的套索、受约束的缩放套索和具有线性等式约束的稀疏 Huber M 估计器。{\sigma} \right) + \lambda \left\lVert \beta\right\rVert_1 \qquad \textrm{subject to} \qquad C\beta = 0 \] 几个凸损失函数 $f(\cdot,\cdot )$。这包括受约束的套索、受约束的缩放套索和具有线性等式约束的稀疏 Huber M 估计器。