Sparse Multivariate Bernoulli Processes in High Dimensions
Sparse Multivariate Bernoulli Processes in High Dimensions
复制标题
高维稀疏多元伯努利过程
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
A. Fletcher
中科院分区：
文献类型：
--
作者：
Parthe Pandit;Mojtaba Sahraee;A. Amini;S. Rangan;A. Fletcher
研究了在高维环境下，当样本数量远小于参数数量时，具有自回归反馈的多元伯努利过程的参数估计问题。这个问题出现在具有尖峰或二值数据的动态系统网络的互连学习中。我们还允许过程依赖于它的过去到一个滞后p，对于一般的$p geq $ 1$，允许在许多应用程序中更真实的建模。在参数张量近似稀疏的假设下，提出并分析了一种$ell_1$正则化极大似然估计。这种估计量的严格分析是具有挑战性的，因为过程的依赖和非高斯性质，以及非线性和多层次反馈的存在。我们根据样本数量、过程的维度、滞后p和模型的其他关键统计特性，推导出均方估计误差的精确上界。所提出的思想可用于其他具有远程相关性的稀疏非线性和非高斯过程的正则化$M$-估计量的严格高维分析。
We consider the problem of estimating the parameters of a multivariate Bernoulli process with auto-regressive feedback in the high-dimensional setting where the number of samples available is much less than the number of parameters. This problem arises in learning interconnections of networks of dynamical systems with spiking or binary valued data. We also allow the process to depend on its past up to a lag p, for a general $p geq 1$, allowing for more realistic modeling in many applications. We propose and analyze an $ell_1$-regularized maximum likelihood (ML) estimator under the assumption that the parameter tensor is approximately sparse. Rigorous analysis of such estimators is made challenging by the dependent and non-Gaussian nature of the process as well as the presence of the nonlinearities and multi-level feedback. We derive precise upper bounds on the mean-squared estimation error in terms of the number of samples, dimensions of the process, the lag $p$ and other key statistical properties of the model. The ideas presented can be used in the rigorous high-dimensional analysis of regularized $M$-estimators for other sparse nonlinear and non-Gaussian processes with long-range dependence.
影响因子：
2.5
作者：
Benjamin Mark;Garvesh Raskutti;R. Willett
通讯作者：
Benjamin Mark;Garvesh Raskutti;R. Willett