混合、包络和层次二元性
Mixtures, envelopes, and hierarchical duality
摘要：
我们开发的混合物和包络表示的目标函数，经常出现在统计之间的连接。我们用“层次对偶”这个术语来指这种联系。“我们的研究结果表明，边缘化和轮廓之间存在一种有趣的、以前未被充分利用的关系，或者说，在凸函数的Fenchel-Moreau定理和拉普拉斯变换的Bernstein-Widder定理之间存在一种等价的关系。我们给出了几组不同的条件下，这样的对偶结果获得。然后，我们扩展现有的工作在几个方面，包括新的概括方差均值模型和多元高斯位置模型的包络表示。这为近端梯度法提供了一种优雅的缺失数据解释，近端梯度法是机器学习中广泛使用的算法。我们展示了几个统计应用，其中所提出的框架导致易于实现的算法，包括一个强大的版本的融合套索，非线性分位数回归通过趋势滤波，和二项式融合双帕累托模型。示例代码可在GitHub上获得，网址为https://github.com/jgscott/hierduals。
英文摘要：
We develop a connection between mixture and envelope representations of objective functions that arise frequently in statistics. We refer to this connection using the term "hierarchical duality." Our results suggest an interesting and previously under-exploited relationship between marginalization and profiling, or equivalently between the Fenchel--Moreau theorem for convex functions and the Bernstein--Widder theorem for Laplace transforms. We give several different sets of conditions under which such a duality result obtains. We then extend existing work on envelope representations in several ways, including novel generalizations to variance-mean models and to multivariate Gaussian location models. This turns out to provide an elegant missing-data interpretation of the proximal gradient method, a widely used algorithm in machine learning. We show several statistical applications in which the proposed framework leads to easily implemented algorithms, including a robust version of the fused lasso, nonlinear quantile regression via trend filtering, and the binomial fused double Pareto model. Code for the examples is available on GitHub at https://github.com/jgscott/hierduals.
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