Sparse additive models are an attractive choice in circumstances calling for modelling flexibility in the face of high dimensionality. We study the signal detection problem and establish the minimax separation rate for the detection of a sparse additive signal. Our result is nonasymptotic and applicable to the general case where the univariate component functions belong to a generic reproducing kernel Hilbert space. Unlike the estimation theory, the minimax separation rate reveals a nontrivial interaction between sparsity and the choice of function space. We also investigate adaptation to sparsity and establish an adaptive testing rate for a generic function space; adaptation is possible in some spaces while others impose an unavoidable cost. Finally, adaptation to both sparsity and smoothness is studied in the setting of Sobolev space, and we correct some existing claims in the literature.
翻译:稀疏加性模型在高维情况下具有建模灵活性,因此备受青睐。我们研究了信号检测问题,并确定了稀疏加性信号检测的极小极大分离速率。我们的结果是非渐近的,并适用于单变量组件函数属于通用的再生核希尔伯特空间的一般情况。与估计理论不同,极小极大分离速率揭示了稀疏性和函数空间选择之间的非平凡交互作用。我们还研究了对稀疏性的自适应性,并为一般的函数空间建立了自适应测试速率;在某些空间中可以实现自适应,而在其他空间中则需要不可避免的代价。最后,我们在Sobolev空间的情况下研究了对稀疏性和平滑性的自适应性,并修正了文献中的一些存在问题的主张。