Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

翻译：混合模型传统上通过将多个分布作为分量相加来表示和学习。允许混合模型减去概率质量或密度，能够显著减少建模复杂分布所需的分量数量。然而，在确保此类减性混合模型仍编码非负函数的前提下学习它们颇具挑战性。我们研究如何通过平方运算来学习深层减性混合模型并执行推理，并在概率电路框架下开展这一工作——该框架使我们能够表示张量化混合模型，并推广其他几种减性模型。我们从理论上证明，允许减法的平方电路类别在表达能力上可超越传统加性混合模型呈指数级增长；并在系列真实分布估计任务中实证展示了这种增强的表达能力。