PHASE TRANSITION IN THE SPIKED RANDOM TENSOR WITH RADEMACHER PRIOR
PHASE TRANSITION IN THE SPIKED RANDOM TENSOR WITH RADEMACHER PRIOR
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DOI：
10.1214/18-aos1763
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发表时间：
2019-10-01
影响因子：
4.5
通讯作者：
Chen, Wei-Kuo
中科院分区：
文献类型：
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作者：
Chen, Wei-Kuo
我们考虑使用从 Rademacher 先验中采样的一阶尖峰来检测对称高斯随机 p 张量 (p >= 3) 的变形的问题。最近，Lesieur 等人。 (Barbier, Krzakala, Macris, Miolane and Zdeborova (2017))，证明存在一个临界阈值 beta(p)，当信噪比超过 beta(p) 时，可以区分尖峰和非尖峰张量，并通过最小均方误差方法弱恢复先验。另一方面，Perry、Wein 和 Bandeira (Perry, Wein and Bandeira (2017)) 证明存在 beta(p)' < beta(p)，使得任何统计假设检验都无法区分这两个张量，即当信噪比小于 beta(p)' 时，它们的总变异距离渐近消失。在这项工作中，我们证明 beta(p) 确实是在总变化距离下严格区分两个张量之间的可区分性和不可区分性的临界阈值。我们的方法基于对具有伊辛自旋的纯 p 自旋模型的高温行为的精细分析，该模型最初源于自旋玻璃领域。特别是，我们将信噪比临界值 beta(p) 确定为伊辛纯 p 自旋平均场自旋玻璃模型的临界温度，以区分高温和低温行为。
We consider the problem of detecting a deformation from a symmetric Gaussian random p-tensor (p >= 3) with a rank-one spike sampled from the Rademacher prior. Recently, in Lesieur et al. (Barbier, Krzakala, Macris, Miolane and Zdeborova (2017)), it was proved that there exists a critical threshold beta(p) so that when the signal-to-noise ratio exceeds beta(p), one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein and Bandeira (Perry, Wein and Bandeira (2017)) proved that there exists a beta(p)' < beta(p) such that any statistical hypothesis test cannot distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than beta(p)'. In this work, we show that beta(p) is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality beta(p) as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure p-spin mean-field spin glass model.