Mutual information in rank-one matrix estimation
Mutual information in rank-one matrix estimation
复制标题
一阶矩阵估计中的互信息
DOI：
10.1109/itw.2016.7606798
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
L. Zdeborová
中科院分区：
文献类型：
--
作者：
Florent Krzakala;Jiaming Xu;L. Zdeborová
我们认为，从噪声和可能性的非线性元素的测量xxT，一个非常通用的问题，包含，例如随机2块模型，子矩阵本地化或随机矩阵的尖峰扰动的知识的n维向量x的估计。使用Guerra [1]提出并随后由Korada和Macris [2]改进的插值方法，我们证明了Bethe互信息（与Bethe自由能相关，并由Lesieur等人推测为精确的。[3]基于非严格腔方法）总是产生精确互信息的上界。使用类似的技术也提供了一个下限。具体来说，我们说明了我们的研究结果的稀疏PCA问题，并观察到（a）我们的边界匹配的参数和（B），有一个相变的区域中的频谱仍然没有信息。虽然我们目前只有秩一对称矩阵估计的情况下，我们的证明技术是很容易扩展到低秩对称矩阵或低秩对称张量估计。
We consider the estimation of a n-dimensional vector x from the knowledge of noisy and possibility non-linear element-wise measurements of xxT, a very generic problem that contains, e.g. stochastic 2-block model, submatrix localization or the spike perturbation of random matrices. Using an interpolation method proposed by Guerra [1] and later refined by Korada and Macris [2], we prove that the Bethe mutual information (related to the Bethe free energy and conjectured to be exact by Lesieur et al. [3] on the basis of the non-rigorous cavity method) always yields an upper bound to the exact mutual information. A lower bound is also provided using a similar technique. For concreteness, we illustrate our findings on the sparse PCA problem, and observe that (a) our bounds match for a large region of parameters and (b) that there exists a phase transition in a region where the spectrum remains uninformative. While we present only the case of rank-one symmetric matrix estimation, our proof technique is readily extendable to low-rank symmetric matrix or low-rank symmetric tensor estimation.
通过消息传递进行子矩阵定位
影响因子：
6
作者：
Hajek, Bruce;Wu, Yihong;Xu, Jiaming
通讯作者：
Xu, Jiaming