AI 中文总结
本文证明总风险暴露约束的均值-方差规划与约束LASSO在任意线性约束下追踪同一条分段线性曲线，并给出两种参数化间的映射及约束拟合的自由度，建立了两个文献领域间的新对应关系。
AI 中文摘要
许多统计程序计算的是整个解路径，而非单一估计量。对于一类程序，该路径是分段线性的，由活动集同伦法从一个角追踪到另一个角。其两个成员完全重合：在$\Sigma=X^\top X$和$\mu=X^top y$下，总风险暴露约束的均值-方差规划与约束LASSO追踪同一条分段线性曲线，并且在任意线性等式和不等式约束下仍然如此。被扫描的参数——无论是杠杆预算还是收益倾斜——并非曲线所注意的选择：在齐次任务下，两种扫描仅相差一个标量。而保持投资组合充分投资的任务则导致径向重新缩放，有效前沿即为该曲线重新缩放后的结果；其拐角未被察觉，因为前沿在这些拐角处连续可微。我们给出了两种参数化之间的映射及其退化的唯一方式。均值-方差选择与LASSO实例化同一个参数二次规划是已有成果（Gärtner, Jaggi and Maria, 2012）。临界线算法本身不在此路径上，本文建立的对应关系是新的。该恒等式是曲线的恒等，而非统计实验的恒等。参数化之间的映射由数据计算得出，因此路径的几何性质得以传递，而响应变量上的平均量则不然。我们有意跨越这一边界，并给出在任意线性约束下约束拟合的自由度。在仅做多且充分投资的情形下，该自由度等于远离边界的持仓期望数量减一。
英文摘要
Many statistical procedures compute an entire solution path rather than a single estimator. For one class the path is piecewise linear, traced corner to corner by an active-set homotopy. Two of its members coincide exactly: under $Σ=X^\top X$ and $μ=X^\top y$ the gross-exposure-constrained mean--variance program and the constrained LASSO trace the same piecewise-linear curve, and they keep doing so under arbitrary linear equality and inequality constraints. Which parameter is swept, a leverage budget or a return tilt, is not a choice the curve notices: under a homogeneous mandate the two sweeps differ by a scalar. A mandate that holds the portfolio invested costs a radial rescaling instead, and the efficient frontier is that curve rescaled; its corners pass unnoticed because the frontier is continuously differentiable where they fall. We give the map between the two parametrisations and the single way it degenerates. That mean--variance selection and the LASSO instantiate one parametric quadratic program is prior art (Gärtner, Jaggi and Maria, 2012). The Critical Line Algorithm itself lies off that route, and the correspondence established here is new. The identity is one of curves, not of statistical experiments. The map between the parametrisations is computed from the data, so the geometry of the path transfers while quantities averaged over the response do not. We cross that boundary deliberately and give the degrees of freedom of the constrained fit under arbitrary linear constraints. In the long-only, fully invested case it is the expected number of holdings away from a bound, less one.
Comments16 pages, 3 figures, 2 tables
