Minimax Estimation of Bandable Precision Matrices
Minimax Estimation of Bandable Precision Matrices
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发表时间：
2017-10
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通讯作者：
Addison Hu;S. Negahban
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作者：
Addison Hu;S. Negahban
逆协方差矩阵为理解多变量环境中的统计模型提供了相当多的见解。特别地，当假设变量的分布是多元正态分布时，逆协方差矩阵（通常称为精度矩阵）中的稀疏模式对应于高斯-马尔可夫图的邻接矩阵表示，其编码变量之间的条件独立性陈述。谱范数下的极大极小结果以前已经建立了协方差矩阵，稀疏和带状，稀疏精度矩阵。我们建立了谱范数下带状精度矩阵的极大极小估计界。我们的研究结果大大提高了现有的界限，特别是，我们发现，估计带状精度矩阵匹配估计带状协方差矩阵的极大极小率。在我们的分析中的关键见解是，我们能够获得几乎没有噪声的估计的精度矩阵的$k \乘以k$子块通过反转略宽块的经验协方差矩阵沿着对角线。我们的理论结果是补充实验证明我们的界限的清晰度。
The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse precision matrices. We establish minimax estimation bounds for estimating banded precision matrices under the spectral norm. Our results greatly improve upon the existing bounds; in particular, we find that the minimax rate for estimating banded precision matrices matches that of estimating banded covariance matrices. The key insight in our analysis is that we are able to obtain barely-noisy estimates of $k \times k$ subblocks of the precision matrix by inverting slightly wider blocks of the empirical covariance matrix along the diagonal. Our theoretical results are complemented by experiments demonstrating the sharpness of our bounds.