We provide a geometric approach to the lasso. We study the tangency of the level sets of the least square objective function with the polyhedral boundary sets $B(t)$ of the parameters in $\mathbb R^p$ with the $\ell_1$ norm equal to $t$. Here $t$ decreases from the value $\hat t$, which corresponds to the actual, nonconstrained minimizer of the least square objective function, denoted by $\hat\beta$. We derive closed exact formulae for the solution of the lasso under the full rank assumption. Our method does not assume iterative numerical procedures and it is, thus, computationally more efficient than the existing algorithms for solving the lasso. We also establish several important general properties of the solutions of the lasso. We prove that each lasso solution form a simple polygonal chain in $\mathbb{R}^p$ with $\hat\beta$ and the origin as the endpoints. There are no two segments of the polygonal chain that are parallel. We prove that such a polygonal chain can intersect interiors of more than one orthant in $\mathbb{R}^p$, but it cannot intersect interiors of more than $p$ orthants, which is, in general, the best possible estimate for non-normalized data. We prove that if a polygonal chain passes from the interior of one to the interior of another orthant, then it never again returns to the interior of the former. The intersection of a chain and the interior of an orthant coincides with a segment minus its end points, which belongs to a ray having $\hat\beta$ as its initial point. We illustrate the results using real data examples as well as especially crafted examples with hypothetical data. Already in $p=2$ case we show a striking difference in the maximal number of quadrants a polygonal chain of a lasso solution can intersect in the case of normalized data, which is $1$ vs. nonnormalized data, which is $2$.

翻译：本文为Lasso问题提供了一种几何方法。我们研究了最小二乘目标函数的等高线与参数空间$\mathbb R^p$中$\ell_1$范数等于$t$的多面体边界集$B(t)$的相切关系。其中$t$从对应无约束最小二乘估计量$\hat\beta$的$\hat t$开始递减。在满秩假设下，我们推导出了Lasso问题的闭式精确解。该方法无需迭代数值计算，因此在计算效率上优于现有Lasso求解算法。我们还建立了Lasso解的若干重要普适性质：证明每个Lasso解在$\mathbb{R}^p$中构成以$\hat\beta$和原点为端点的简单多边形链；该多边形链不存在平行线段；证明该链可能穿过多个象限内部，但最多只能穿过$p$个象限——对于非标准化数据而言这是最优上界；证明当多边形链从一个象限内部穿入另一象限内部后，绝不会再次返回原象限；链与象限内部的交点构成以$\hat\beta$为起点的射线上的开线段。我们通过实际数据案例与精心构建的假设数据示例验证了结论。即使在$p=2$情形中，我们也揭示了标准化数据（最多穿过1个象限）与非标准化数据（最多穿过2个象限）情况下Lasso解多边形链可穿越象限最大数量的显著差异。