Dimensionality increase for error correction in the interaction between information space and the physical world
Tatyana Barron
TL;DR
Problem addressed: how to realize error-correction and solution-finding when the physical world and information space are naturally infinite-dimensional and can be modeled as submanifolds of a common Hilbert space $H$. Approach: formalize a pair of submanifolds $S\subset W=H$ and $E\subset H$, with mappings between them, and prove an existence theorem that deforms a given submanifold $A\subset S$ to a target submanifold $B\subset W$ via a higher-dimensional ambient submanifold. Key contribution: Theorem 1 guarantees a submanifold $X$ with $n\le \dim X\le 2n+1$ and a smooth family $Y(t)$ such that $Y(a)=A$ and $Y(b)=B$, providing an existence principle for a class of problems. Significance: frames error correction and dimensionality augmentation within a geometric, Hilbert-space setting, with ties to digital twins, world models, and potential extensions to fuzzy or quantum information processing.
Abstract
The evolution of human intelligence led to the huge amount of data in the information space. Accessing and processing this data helps in finding solutions to applied problems based on finite-dimensional models. We argue, that formally, such a mathematical model can be embedded into a higher-dimensional model inside of which a desired solution will exist. In our model, the physical world and the information space are submanifolds of infinite-dimensional Hilbert spaces, and the processes, including information transmission, are maps between the submanifolds of the physical world or of the information space. We discuss how our perspective fits in the context of existing literature. Our theorem states that a submanifold in the parameter space of the physical world can be deformed to a target submanifold outside that space, with an appropriate count of the deformation parameters. We interpret this assertion as an existence result for a class of problems and we discuss further steps.