关于旋转傅里叶变换之间的点积的短文
A short letter on the dot product between rotated Fourier transforms
摘要：
空间语义指针（SSP）最近成为表示和转换连续空间的强大工具，在认知建模和深度学习中有许多应用。SSP的基础是表示$n$维空间中不同点的向量之间的“相似性”的概念--通常是傅立叶域中具有旋转单位长度复系数的向量之间的点积或余弦相似性。相似性度量先前已被证明是欧几里德距离的高斯函数。与此猜想相反，我们推导出一个简单的三角公式，空间位移的相似性，并证明，在傅立叶系数是均匀的i.i.d.的情况下，期望的相似性是归一化sinc函数的乘积：$\prod_{k=1}^{n} \operatorname{sinc} \left( a_k \right)$，其中$\mathbf{a} \in \mathbb{R}^n$是两个$n$维点之间的空间位移。这在空间和SSP的相似性之间建立了直接联系，这反过来又有助于支持一个有用的数学框架，用于构建操纵空间结构的神经网络。
英文摘要：
Spatial Semantic Pointers (SSPs) have recently emerged as a powerful tool for representing and transforming continuous space, with numerous applications to cognitive modelling and deep learning. Fundamental to SSPs is the notion of "similarity" between vectors representing different points in $n$-dimensional space -- typically the dot product or cosine similarity between vectors with rotated unit-length complex coefficients in the Fourier domain. The similarity measure has previously been conjectured to be a Gaussian function of Euclidean distance. Contrary to this conjecture, we derive a simple trigonometric formula relating spatial displacement to similarity, and prove that, in the case where the Fourier coefficients are uniform i.i.d., the expected similarity is a product of normalized sinc functions: $\prod_{k=1}^{n} \operatorname{sinc} \left( a_k \right)$, where $\mathbf{a} \in \mathbb{R}^n$ is the spatial displacement between the two $n$-dimensional points. This establishes a direct link between space and the similarity of SSPs, which in turn helps bolster a useful mathematical framework for architecting neural networks that manipulate spatial structures.
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