When Does a Mixture of Products Contain a Product of Mixtures?

为什么一个乘积的混合包含了某个混合的乘积？（搞不懂）

Abstract

We deriverelations between theoretical properties of restricted Boltzmann machines(RBMs), popular machine learning models which form the building blocks of deeplearning models, and several natural notions from discrete mathematics andconvex geometry. We give implications and equivalences relatingRBM-representable probability distributions, perfectly reconstructible inputs,Hamming modes, zonotopes and zonosets, point congurations in hyperplanearrangements, linear threshold codes, and multi-covering numbers of hypercubes.As a motivating application, we prove results on the relative representationalpower of mixtures of product distributions and products of mixtures of pairs ofproduct distributions (RBMs) that formally justify widely held intuitions aboutdistributed representations. In particular, we show that a mixture of productsrequiring an exponentially larger number of parameters is needed to representthe probability distributions which can be obtained as products of mixtures.

我们得到了限制Boltzman机（RBMs）（一个形成了深度学习模型的构件模块的流行的机器学习模型）和一些来自离散数学和凸几何的自然提法之间的关系。我们给出了和RBM-代表性的概率分布相关的含义和等价，完美的重建输入，加重平均Hamming模式，zonotopes 和zonosets，超plane安排中的点设置，线性阈值代码，以及超方的多-覆盖数。作为一个启发应用，我们证明了乘积分布混合的有关联的代表性的幂与乘积分布的对子的混合积（RBMs，正式地证明广泛地用作关于分布式表示上的直觉印象）的结果。特别地，我们证明乘积的混合需要一个指数级别的参数来表示概率分布，这可以通过混合乘积得到。