稀疏张量PCA中的全有或全无现象
The All-or-Nothing Phenomenon in Sparse Tensor PCA
摘要：
我们研究了估计一个被加性高斯噪声破坏的秩为1的稀疏张量的统计问题，该模型也被称为稀疏张量PCA。我们发现，伯努利和伯努利-Rademacher分布的信号和\emph{for all}稀疏水平是次线性的信号的维度，稀疏张量PCA模型表现出相变称为\emph{all-or-nothing phenomenon}。这是这样的性质，即对于某个信噪比（SNR）$\mathrm{SNR_c}$和任何固定的$ε>0$，如果模型的SNR低于$\left(1-ε\right)\mathrm{SNR_c}$，则不可能实现与隐藏信号的任何任意小的恒定相关，而如果SNR高于$\left(1+ε\right)\mathrm{SNR_c}$，则可以实现与隐藏信号的几乎完美的相关。全有或全无现象最初是在稀疏线性回归的背景下建立的，在过去的一年中，稀疏2张量（矩阵）PCA，伯努利组测试和广义线性模型也是如此。我们的结果遵循一个更一般的结果表明，对于任何高斯加性模型与离散均匀先验，全有或全无的现象如下的先验分布的支持的一个适当定义的“近正交”属性的直接结果。
英文摘要：
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.
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