论文标题
稀疏的尖峰矩阵估计中的全部或全无的统计和计算相变
All-or-nothing statistical and computational phase transitions in sparse spiked matrix estimation
论文作者
论文摘要
我们确定统计和计算限制的估计，以估算由加性高斯噪声矩阵损坏的排名矩阵（尖峰），在稀疏的极限中，基础的隐藏矢量（构造等级单矩阵）具有许多非零组件，可缩放量度与信号的速度限制，并适用于信号的速度，并具有适当的速度。我们证明，对于尖峰和观察到的嘈杂矩阵之间的渐近互信息，我们证明了显式的低维变异公式，并分析了稀疏状态中传递算法的近似消息。对于Bernoulli和Bernoulli-Rademacher分布的向量，当稀疏性和信号强度满足适当的缩放关系时，我们发现渐近最小和算法的均值均值误差的全部或全部相变。这些从最大可能的值跳到零，在定义明确的信号到噪声阈值下，我们确切确定其渐近值。在渐近制度中，统计到偏我们的差距有分歧，表明稀疏恢复很难在近似消息传中传递。
We determine statistical and computational limits for estimation of a rank-one matrix (the spike) corrupted by an additive gaussian noise matrix, in a sparse limit, where the underlying hidden vector (that constructs the rank-one matrix) has a number of non-zero components that scales sub-linearly with the total dimension of the vector, and the signal-to-noise ratio tends to infinity at an appropriate speed. We prove explicit low-dimensional variational formulas for the asymptotic mutual information between the spike and the observed noisy matrix and analyze the approximate message passing algorithm in the sparse regime. For Bernoulli and Bernoulli-Rademacher distributed vectors, and when the sparsity and signal strength satisfy an appropriate scaling relation, we find all-or-nothing phase transitions for the asymptotic minimum and algorithmic mean-square errors. These jump from their maximum possible value to zero, at well defined signal-to-noise thresholds whose asymptotic values we determine exactly. In the asymptotic regime the statistical-to-algorithmic gap diverges indicating that sparse recovery is hard for approximate message passing.
