平均退火马尔可夫链达到张量主成分分析的算法阈值，第一部分： glauber动力学
Averaged Annealed Markov Chains Achieve the Algorithmic Threshold for Tensor PCA, Part I: Glauber Dynamics
AI总结：
该研究针对张量PCA问题，证明辅以退火和遍历平均的Glauber动力学，可从均匀随机初始化出发，在算法阈值以上实现多项式时间强恢复，打破了局部马尔可夫链劣于全局算法的认知。
摘要：
在张量主成分分析（tensor PCA）问题中，我们需要从固定k≥3的第k阶张量幂的噪声观测中恢复{±1}^N中的未知向量。张量PCA存在已被广泛研究的信息-计算鸿沟：谱方法、平方和（sum-of-squares）等全局算法在算法阈值以上有效，而在该阈值以下，我们已知针对这些算法及其他受限计算模型的匹配下界。相比之下，马尔可夫链等局部算法此前仅被知晓能在多项式更大的信噪比下从随机初始化成功，且被认为在张量PCA中劣于全局算法。我们证明，当Glauber动力学辅以退火和遍历平均时，可在多项式时间内从均匀随机初始化出发，在算法阈值以上实现强恢复，表明局部马尔可夫链能从无信息的冷启动一直成功到算法阈值。
英文摘要：
In the tensor principal components analysis (tensor PCA) problem, we are asked to recover an unknown vector in $\{\pm1\}^N$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. We show that Glauber dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization, demonstrating that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.
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