We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can hypothesis test between high-dimensional probability measures. These functions are a generalization, proposed in Hopkins' 2018 thesis but seldom studied since, of low degree polynomials (LDP), a class widely used in recent literature as a proxy for all efficient algorithms for tasks in statistics and optimization. Instead of the orthogonal polynomial decompositions used in LDP calculations, our analysis of LCDF is based on the Efron-Stein or ANOVA decomposition, making it much more broadly applicable. By way of illustration, we prove channel universality for the success of LCDF in testing for the presence of sufficiently "dilute" random signals through noisy channels: the efficacy of LCDF depends on the channel only through the scalar Fisher information for a class of channels including nearly arbitrary additive i.i.d. noise and nearly arbitrary exponential families. As applications, we extend lower bounds against LDP for spiked matrix and tensor models under additive Gaussian noise to lower bounds against LCDF under general noisy channels. We also give a simple and unified treatment of the effect of censoring models by erasing observations at random and of quantizing models by taking the sign of the observations. These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.

翻译：我们研究了低坐标度函数（LCDF）——即依赖于向量中少数条目子集的函数的线性组合——能否在高维概率测度之间进行假设检验。这类函数是低度多项式（LDP）的推广，由Hopkins在2018年的论文中提出但此后鲜有研究，而LDP在近期文献中被广泛用作统计与优化任务中所有高效算法的代理。与LDP计算中使用的正交多项式分解不同，我们对LCDF的分析基于Efron-Stein分解或ANOVA分解，使其具有更广泛的适用性。作为示例，我们证明了通过含噪信道检测是否存有足够“稀疏”随机信号时，LCDF成功的信道普适性：对于包括近乎任意加性独立同分布噪声和近乎任意指数族在内的信道类别，LCDF的有效性仅通过标量Fisher信息依赖于信道。在应用中，我们将加性高斯噪声下尖峰矩阵模型和张量模型的LDP下界推广为一般含噪信道下的LCDF下界。我们还对通过随机擦除观测的删失模型和通过取观测符号的量化模型的影响给出了简洁统一的处理。这些结果是针对这些模型在信道非少数特例时首个针对任何大规模算法类的计算下界，从而为若干统计-计算鸿沟的普适性提供了首个实质性证据。