学习具有记忆的高维马尔可夫过程的影响图
Learning the Influence Graph of a High-Dimensional Markov Process with Memory
AI总结：
针对具有记忆的高维离散时间马尔可夫过程，论文扩展i.i.d.图模型学习算法，仅用二进制观测即可在度有界条件下以对数样本恢复有向影响图，并通过平稳分布收敛速率界分析样本复杂度。
摘要：
受社交网络、神经系统和金融风险分析中多种应用的启发，我们考虑学习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）影响图或因果图的问题。在任意离散时刻，该多变量过程的每个观测变量都是一个随机长度的二进制字符串，由一个不可观测或隐藏的、取值于[0,1]的标量参数化。与各变量对应的隐藏标量按照由底层影响图决定的离散时间线性随机动力学演化，该图的节点就是这些变量。我们将一种用于学习i.i.d.图模型的现有算法扩展到这种具有记忆的马尔可夫环境，并证明在影响图的度有界时，该算法能够利用二进制观测学习影响图，所需样本数（相对于变量或节点数量）为对数级。本工作的关键分析贡献在于，通过依据影响图参数对具有记忆的观测马尔可夫过程收敛到其平稳分布的收敛速率进行上界和下界界定，推导出样本复杂度结果。
英文摘要：
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.
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