Adapting to unknown sparsity by controlling the false discovery rate
通过控制错误发现率适应未知稀疏性AI 解读 ↓
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通过控制错误发现率适应未知稀疏性
- 研究问题
- 在白噪声中恢复稀疏程度未知的高维向量时，错误发现率驱动的阈值能否同时适应不同稀疏类与损失函数？
- 方法与思路
- 作者以非零比例、排序分量的幂律界及小阶范数定义稀疏性，构造FDR驱动的自适应阈值，并分析风险和复杂度惩罚联系。
- 材料陈述的贡献
- 摘要称，FDR参数极限不超过二分之一并满足额外下界时，可获得精确渐近极小极大性；更大的极限会增加风险比。
- 阅读关注点
- 核对噪声模型、稀疏类及损失范围，区分FDR保证与估计风险最优性，并检查控制参数序列的全部条件和风险膨胀界。
- 阅读边界
- 仅据版本未注明的OpenAlex摘要；不能脱离白噪声及参数空间条件，将二分之一界限推广为所有多重检验的最优设置。
原文摘要
We attempt to recover an n-dimensional vector observed in white noise, where n is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the ℓp norm for p small. We obtain a procedure which is asymptotically minimax for ℓr loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the false discovery rate (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain expected fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter qn also plays a determining role in asymptotic minimaxity. If q=lim qn∈[0,1/2] and also qn>γ/log(n), we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, q=lim qn∈(1/2,1] forces the risk to exceed the minimax risk by a factor growing with q. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form 2⋅log(potential model size/actual model sizes). We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.