置换不变高斯矩阵模型
Permutation Invariant Gaussian Matrix Models
摘要：
排列不变高斯矩阵模型是近年来发展起来的一种计算语言学模型。求解了5参数模型族。本文用表示论方法求解了一般的13参数高斯模型，该模型可以看作是一个零维的量子场论。我们表示的两个线性和十一个二次项的行动表示论参数。这些参数是简单的二次表达式的系数，根据矩阵变量的适当线性组合，变换成对称群$S_D$的特定不可约表示，其中$D$是矩阵的大小。它们允许识别的约束，确保收敛的高斯措施和明确定义的期望值的多项式函数的随机矩阵在所有订单。一个图论的解释是已知的，允许在线性，二次和更高的订单矩阵的置换不变量的枚举。我们表示的期望值的所有二次图基不变量和选择的三次和四次不变量的表示理论参数的模型。
英文摘要：
Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group $S_D$ where $D$ is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.
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