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：稀疏加性模型在面对高维数据需要建模灵活性时是一个有吸引力的选择。我们研究信号检测问题，并建立了检测稀疏加性信号的极小极大分离速率。我们的结果是非渐近的，适用于单变量分量函数属于一般再生核希尔伯特空间的一般情形。与估计理论不同，极小极大分离速率揭示了稀疏性与函数空间选择之间非平凡的相互作用。我们还研究了针对稀疏性的自适应问题，并为一般函数空间建立了自适应检验速率；在某些空间中可以实现自适应，而在其他空间中则不可避免地需要付出代价。最后，我们在索伯列夫空间背景下研究了针对稀疏性和光滑性的自适应问题，并纠正了文献中一些现有的论断。