Inner products of neural network feature maps arises in a wide variety of machine learning frameworks as a method of modeling relations between inputs. This work studies the approximation properties of inner products of neural networks. It is shown that the inner product of a multi-layer perceptron with itself is a universal approximator for symmetric positive-definite relation functions. In the case of asymmetric relation functions, it is shown that the inner product of two different multi-layer perceptrons is a universal approximator. In both cases, a bound is obtained on the number of neurons required to achieve a given accuracy of approximation. In the symmetric case, the function class can be identified with kernels of reproducing kernel Hilbert spaces, whereas in the asymmetric case the function class can be identified with kernels of reproducing kernel Banach spaces. Finally, these approximation results are applied to analyzing the attention mechanism underlying Transformers, showing that any retrieval mechanism defined by an abstract preorder can be approximated by attention through its inner product relations. This result uses the Debreu representation theorem in economics to represent preference relations in terms of utility functions.

通过分析内积关系，研究了神经网络特征图的内积在模拟输入之间的关系方面的广泛应用，证明了多层感知器自身的内积是对称正定关系函数的普遍逼近器，而两个不同多层感知器的内积是非对称关系函数的普遍逼近器，并通过内积关系将任何抽象预订定义的检索机制近似为注意力机制，从而应用于分析Transformer背后的注意力机制。同时，利用经济学中的Debreu表示定理以效用函数的形式表示偏好关系。

关系函数和注意力机制的近似