The convergence of expectation-maximization (EM)-based algorithms typically requires continuity of the likelihood function with respect to all the unknown parameters (optimization variables). The requirement is not met when parameters comprise both discrete and continuous variables, making the convergence analysis nontrivial. This paper introduces a set of conditions that ensure the convergence of a specific class of EM algorithms that estimate a mixture of discrete and continuous parameters. Our results offer a new analysis technique for iterative algorithms that solve mixed-integer non-linear optimization problems. As a concrete example, we prove the convergence of the EM-based sparse Bayesian learning algorithm in [1] that estimates the state of a linear dynamical system with jointly sparse inputs and bursty missing observations. Our results establish that the algorithm in [1] converges to the set of stationary points of the maximum likelihood cost with respect to the continuous optimization variables.

翻译：基于期望最大化（EM）的算法的收敛性通常要求似然函数关于所有未知参数（优化变量）连续。当参数同时包含离散变量和连续变量时，该条件不满足，使得收敛性分析变得非平凡。本文引入了一组条件，确保一类估计离散与连续参数混合的特定EM算法的收敛性。我们的结果为求解混合整数非线性优化问题的迭代算法提供了一种新的分析技术。作为一个具体实例，我们证明了[1]中基于EM的稀疏贝叶斯学习算法的收敛性，该算法估计具有联合稀疏输入和突发性缺失观测的线性动态系统状态。我们的结果确立了[1]中的算法收敛到关于连续优化变量的最大似然代价的驻点集。