In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products. We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative.

翻译：在本文中, 我们在随机矢量的集合中考虑内部产品的低度多元值。 当随机矢量为高斯、 球体或布林值时, 我们给多元矢量的矢量空间提供了几乎正反基础 。 在全部三种情况下, 我们的基础都承认根据内产物底图的地形学, 一个有趣的组合式描述 。 我们还分析了我们基础中两种多元值产品的预期值 。 在全部三种情况下, 我们显示这一预期值可以表现为内部产品底图上的匹配值的集合 。 在高斯和布林案例中, 我们显示这一预期值总是非负值 。 在球中, 我们显示这一预期值可能是负值, 但是我们推测, 如果内产物底图是平坦的, 这个预期值将永远是非负值 。