We present a unified approach for learning the parameters of Sum-Product
networks (SPNs). We prove that any complete and decomposable SPN is equivalent
to a mixture of trees where each tree corresponds to a product of univariate
distributions. Based on the mixture model perspective, we characterize the
objective function when learning SPNs based on the maximum likelihood
estimation (MLE) principle and show that the optimization problem can be
formulated as a signomial program. We construct two parameter learning
algorithms for SPNs by using sequential monomial approximations (SMA) and the
concave-convex procedure (CCCP), respectively. The two proposed methods
naturally admit multiplicative updates, hence effectively avoiding the
projection operation. With the help of the unified framework, we also show
that, in the case of SPNs, CCCP leads to the same algorithm as Expectation
Maximization (EM) despite the fact that they are different in general.

本文提出了一种统一的方法来学习 Sum-Product networks (SPNs) 的参数，在混合模型的角度下，我们表征了基于最大似然估计 (MLE) 原理的学习 SPNs 的目标函数，并展示了优化问题可以被转化为一个符号函数规划的形式。同时，我们使用顺序单项逼近（SMA）和凸凹过程（CCCP）构建了两种 SPNs 参数学习算法，这些方法自然地采用了乘法更新，从而有效地避免了投影操作。通过统一的框架，我们还展示，在 SPNs 的情况下，CCCP 导致与期望最大化（Expectation  Maximization，EM）相同的算法尽管它们在一般情况下是不同的。