This work performs a non-asymptotic analysis of the generalized Lasso under the assumption of sub-exponential data. Our main results continue recent research on the benchmark case of (sub-)Gaussian sample distributions and thereby explore what conclusions are still valid when going beyond. While many statistical features remain unaffected (e.g., consistency and error decay rates), the key difference becomes manifested in how the complexity of the hypothesis set is measured. It turns out that the estimation error can be controlled by means of two complexity parameters that arise naturally from a generic-chaining-based proof strategy. The output model can be non-realizable, while the only requirement for the input vector is a generic concentration inequality of Bernstein-type, which can be implemented for a variety of sub-exponential distributions. This abstract approach allows us to reproduce, unify, and extend previously known guarantees for the generalized Lasso. In particular, we present applications to semi-parametric output models and phase retrieval via the lifted Lasso. Moreover, our findings are discussed in the context of sparse recovery and high-dimensional estimation problems.
翻译:这项工作对假设的亚穷度数据假设的普惠性Lasso进行非抽取性分析。 我们的主要结果继续最近对(子)Gaussian样本分布的基准案例进行的研究,从而探索在超出范围时哪些结论仍然有效。 虽然许多统计特征仍然不受影响(例如一致性和误差衰减率),但关键差异表现在如何衡量假设的复杂性上。结果显示,估计错误可以通过两个复杂参数来控制,这些参数自然产生于基于通用链的验证战略。产出模型可能是无法实现的,而输入矢量的唯一要求是伯斯坦型的通用集中不平等,可以用于各种次扩展分布。这种抽象方法使我们能够复制、统一和扩展以前已知的普遍激光索的保证。特别是,我们提出了半参数输出模型的应用,并通过解除的激光索进行阶段的检索。此外,我们的调查结果是在零星恢复和高维度估计问题的背景下讨论的。