In general, objects can be distinguished on the basis of their features, such as color or shape. In particular, it is assumed that similarity judgments about such features can be processed independently in different metric spaces. However, the unsupervised categorization mechanism of metric spaces corresponding to object features remains unknown. Here, we show that the artificial neural network system can autonomously categorize metric spaces through representation learning to satisfy the algebraic independence between neural networks, and project sensory information onto multiple high-dimensional metric spaces to independently evaluate the differences and similarities between features. Conventional methods often constrain the axes of the latent space to be mutually independent or orthogonal. However, the independent axes are not suitable for categorizing metric spaces. High-dimensional metric spaces that are independent of each other are not uniquely determined by the mutually independent axes, because any combination of independent axes can form mutually independent spaces. In other words, the mutually independent axes cannot be used to naturally categorize different feature spaces, such as color space and shape space. Therefore, constraining the axes to be mutually independent makes it difficult to categorize high-dimensional metric spaces. To overcome this problem, we developed a method to constrain only the spaces to be mutually independent and not the composed axes to be independent. Our theory provides general conditions for the unsupervised categorization of independent metric spaces, thus advancing the mathematical theory of functional differentiation of neural networks.

翻译：一般而言，物体可以根据其特征（如颜色或形状）进行区分。特别地，我们假设关于此类特征的相似性判断可以在不同的度量空间中独立处理。然而，与物体特征相对应的度量空间的无监督分类机制仍然未知。本文中，我们证明人工神经网络系统能够通过表征学习自主地对度量空间进行分类，以满足神经网络之间的代数独立性，并将感官信息投影到多个高维度量空间中以独立评估特征间的差异与相似性。传统方法通常约束潜在空间的坐标轴相互独立或正交。然而，独立坐标轴并不适用于度量空间的分类。相互独立的高维度量空间并非由相互独立的坐标轴唯一确定，因为任意独立坐标轴的组合均可构成相互独立的空间。换言之，相互独立的坐标轴无法自然地分类不同的特征空间（如颜色空间与形状空间）。因此，约束坐标轴相互独立会导致高维度量空间难以分类。为解决这一问题，我们开发了一种方法：仅约束空间相互独立，而不要求构成空间的坐标轴独立。我们的理论为独立度量空间的无监督分类提供了普适条件，从而推进了神经网络功能分化的数学理论。