Motivated by multiple applications in social networks, nervous systems, and financial risk analysis, we consider the problem of learning the underlying (directed) influence graph or causal graph of a high-dimensional multivariate discrete-time Markov process with memory. At any discrete time instant, each observed variable of the multivariate process is a binary string of random length, which is parameterized by an unobservable or hidden [0,1]-valued scalar. The hidden scalars corresponding to the variables evolve according to discrete-time linear stochastic dynamics dictated by the underlying influence graph whose nodes are the variables. We extend an existing algorithm for learning i.i.d. graphical models to this Markovian setting with memory and prove that it can learn the influence graph based on the binary observations using logarithmic (in number of variables or nodes) samples when the degree of the influence graph is bounded. The crucial analytical contribution of this work is the derivation of the sample complexity result by upper and lower bounding the rate of convergence of the observed Markov process with memory to its stationary distribution in terms of the parameters of the influence graph.

翻译：受社交网络、神经系统和金融风险分析中多种应用的启发，我们研究如何学习一个具有记忆的高维多变量离散时间马尔可夫过程的底层（有向）影响图或因果图。在每个离散时间点，该多变量过程的每个观测变量均为随机长度的二进制字符串，其参数由一个不可观测或隐藏的[0,1]值标量所参数化。这些对应于变量的隐藏标量，按照由底层影响图（其节点即为变量）所决定的离散时间线性随机动力学演化。我们将一种现有的独立同分布图模型学习算法扩展至这一具有记忆的马尔可夫场景，并证明当影响图的度有界时，该算法能够基于二进制观测值，使用对数级（关于变量或节点数量）的样本学习影响图。本工作的关键分析贡献在于，通过用影响图的参数界定所观测的带记忆马尔可夫过程收敛到其平稳分布的速率上下界，推导出了样本复杂度结果。