控制变换稀疏性中的错误发现率：分裂Knockoff方法
Controlling the False Discovery Rate in Transformational Sparsity: Split Knockoffs
AI总结：
本文提出分裂Knockoff方法，用于在变换稀疏性设置中控制错误发现率，通过变量与数据分裂及逆超鞅结构实现可证明的FDR控制，并在模拟和ADNI数据上验证了效果。
摘要：
在变量选择过程中控制错误发现率（FDR）对于可复现的发现至关重要，并且这一课题在稀疏线性模型中已被广泛研究。然而，在稀疏性约束并非直接施加于参数本身，而是施加于待估计参数的线性变换上的场景中，这一问题在很大程度上仍未解决。此类场景的例子包括全变差、小波变换、融合LASSO和趋势滤波。在本文中，我们针对这种变换稀疏性设置，提出了一种数据自适应的FDR控制方法，称为分裂Knockoff方法。所提出的方法同时利用了变量分裂和数据分裂。线性变换约束被放宽为在提升参数空间中的欧几里得邻近约束，从而产生一个正交设计，使得正交分裂Knockoff构造成为可能。为了克服由于变换带来的异质噪声而导致的可交换性失效的挑战，我们通过数据分裂开发了新的逆超鞅结构，以实现可证明的FDR控制而不牺牲功效。模拟实验表明，所提出的方法能够达到期望的FDR和功效。我们还将其应用于阿尔茨海默病研究，基于结构磁共振成像数据集（ADNI）可以发现萎缩的脑区域及其异常连接。
英文摘要：
Controlling the False Discovery Rate (FDR) in a variable selection procedure is critical for reproducible discoveries, and it has been extensively studied in sparse linear models. However, it remains largely open in scenarios where the sparsity constraint is not directly imposed on the parameters but on a linear transformation of the parameters to be estimated. Examples of such scenarios include total variations, wavelet transforms, fused LASSO, and trend filtering. In this paper, we propose a data-adaptive FDR control method, called the Split Knockoff method, for this transformational sparsity setting. The proposed method exploits both variable and data splitting. The linear transformation constraint is relaxed to its Euclidean proximity in a lifted parameter space, which yields an orthogonal design that enables the orthogonal Split Knockoff construction. To overcome the challenge that exchangeability fails due to the heterogeneous noise brought by the transformation, new inverse supermartingale structures are developed via data splitting for provable FDR control without sacrificing power. Simulation experiments demonstrate that the proposed methodology achieves the desired FDR and power. We also provide an application to Alzheimer's Disease study, where atrophy brain regions and their abnormal connections can be discovered based on a structural Magnetic Resonance Imaging dataset (ADNI).
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