Unsupervised categorization of similarity measures

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.

An algebraic theory to discriminate qualia in the brain

The mind-brain problem is to bridge relations between in higher mental events and in lower neural events. To address this, some mathematical models have been proposed to explain how the brain can represent the discriminative structure of qualia, but they remain unresolved due to a lack of validation methods. To understand the qualia discrimination mechanism, we need to ask how the brain autonomously develops such a mathematical structure using the constructive approach. Here we show that a brain model that learns to satisfy an algebraic independence between neural networks separates metric spaces corresponding to qualia types. We formulate the algebraic independence to link it to the other-qualia-type invariant transformation, a familiar formulation of the permanence of perception. The learning of algebraic independence proposed here explains downward causation, i.e. the macro-level relationship has the causal power over its components, because algebra is the macro-level relationship that is irreducible to a law of neurons, and a self-evaluation of algebra is used to control neurons. The downward causation is required to explain a causal role of mental events on neural events, suggesting that learning algebraic structure between neural networks can contribute to the further development of a mathematical theory of consciousness.