论文标题
稀疏张量PCA中的全有或无所不在的现象
The All-or-Nothing Phenomenon in Sparse Tensor PCA
论文作者
论文摘要
我们研究了估计一个被添加剂高斯噪声损坏的排名较差张量的统计问题，该模型也称为稀疏张量PCA。我们表明，对于Bernoulli和Bernoulli-Rademacher分布的信号，以及\ emph {对于在信号尺寸中具有sublerear的所有}稀疏级别，稀疏的张量PCA模型表现出一个相位过渡，称为\ emph {全户外现象}。这是某种信噪比（SNR）$ \ MATHRM {snr_c} $和任何固定$ε> 0 $的属性，如果该模型的SNR低于$ \ left（1-ε\ right）\ Mathrm {snr_c} $ $ \ left（1+ε\ right）\ mathrm {snr_c} $，那么可以与隐藏信号实现几乎完美的相关性。最初是在稀疏线性回归的背景下建立的全或全部现象，并且在去年也是在稀疏的2汤匙（Matrix）PCA，Bernoulli组测试和广义线性模型的背景下。我们的结果取决于更一般的结果表明，对于任何具有离散统一先验的高斯添加剂模型，全或全无的现象遵循了先前分布支持的适当定义的“接近正交性”属性的直接结果。
We study the statistical problem of estimating a rank-one sparse tensor corrupted by additive Gaussian noise, a model also known as sparse tensor PCA. We show that for Bernoulli and Bernoulli-Rademacher distributed signals and \emph{for all} sparsity levels which are sublinear in the dimension of the signal, the sparse tensor PCA model exhibits a phase transition called the \emph{all-or-nothing phenomenon}. This is the property that for some signal-to-noise ratio (SNR) $\mathrm{SNR_c}$ and any fixed $ε>0$, if the SNR of the model is below $\left(1-ε\right)\mathrm{SNR_c}$, then it is impossible to achieve any arbitrarily small constant correlation with the hidden signal, while if the SNR is above $\left(1+ε\right)\mathrm{SNR_c}$, then it is possible to achieve almost perfect correlation with the hidden signal. The all-or-nothing phenomenon was initially established in the context of sparse linear regression, and over the last year also in the context of sparse 2-tensor (matrix) PCA, Bernoulli group testing, and generalized linear models. Our results follow from a more general result showing that for any Gaussian additive model with a discrete uniform prior, the all-or-nothing phenomenon follows as a direct outcome of an appropriately defined "near-orthogonality" property of the support of the prior distribution.
