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Dynamic Geometry: open a new era of next-generation AI driven by mathematics [pdf]

2 points by wchswchs 1y ago

wchswchs 1y ago
The remarkable successes of contemporary artificial intelligence (AI) have been largely driven by empiricism, but the lack of a solid theoretical foundation has produced deep crises in interpretability, safety, and generalization. This paper proposes a new mathematical framework called ”Dynamic Geometry,” aimed at establishing a rigorous axiomatic basis for AI. The framework reconceptualizes AI systems as parameter-driven multilayer families of holistic operators on Banach spaces, whose core dynamics are determined by the evolution of operator spectra — the so-called ”spectral flow.” By formalizing representation learning as spectral projection, relating interpretability to algebraic-topological invariants (such as Ktheory and cyclic homology), linking the realization of “infinite context” with Lyapunov stability of long-time evolution and tying safety (e.g., hallucination control) to verifiable mathematical evidence, Dynamic Geometry provides a fundamental path to constructing provable, robust, and general AI systems. We prove that this framework can encompass any finite-parameter deep learning model in terms of expressive power, and through an analysis based on spectral stability we reveal conditional advantages in its generalization behavior. Dynamic Geometry not only offers a unified theoretical framework to address the current crises in AI, but may also open a new era of next-generation AI driven by first principles and rigorously defined mathematics.
wchswchs 1y ago
History repeats itself — dynamic geometry is undergoing the same kind of development that the Turing machine once experienced: 1. The Turing machine was initially a theoretical success but physically impossible to build at the time. Dynamic geometry likewise has a self-consistent, rigorous mathematical theory, yet its numerical/computational costs are very large. 2. The Turing machine is a “finite-state machine” acting on an “infinite tape.” Dynamic geometry is an abstraction that uses “finite parameters” to represent an “infinite-dimensional” space. Both involve a move from the finite to the infinite. The Turing machine later became the theoretical foundation of the von Neumann architecture (the universal computer). Dynamic geometry needs an equivalent von Neumann — an architecture for AGI. This is the direction I am working toward: to realize a “von Neumann” moment for AI. When that moment arrives, the Turing Award will be virtually guaranteed. If the Turing machine made machines computable, then dynamic geometry makes intelligence computable. AGI would no longer be a qualitative or philosophical claim made by leading figures, it would be something that can be defined and computed mathematically.

2 points by wchswchs 1y ago

The remarkable successes of contemporary artificial intelligence (AI) have been largely driven by empiricism, but the lack of a solid theoretical foundation has produced deep crises in interpretability, safety, and generalization. This paper proposes a new mathematical framework called ”Dynamic Geometry,” aimed at establishing a rigorous axiomatic basis for AI. The framework reconceptualizes AI systems as parameter-driven multilayer families of holistic operators on Banach spaces, whose core dynamics are determined by the evolution of operator spectra — the so-called ”spectral flow.” By formalizing representation learning as spectral projection, relating interpretability to algebraic-topological invariants (such as Ktheory and cyclic homology), linking the realization of “infinite context” with Lyapunov stability of long-time evolution and tying safety (e.g., hallucination control) to verifiable mathematical evidence, Dynamic Geometry provides a fundamental path to constructing provable, robust, and general AI systems. We prove that this framework can encompass any finite-parameter deep learning model in terms of expressive power, and through an analysis based on spectral stability we reveal conditional advantages in its generalization behavior. Dynamic Geometry not only offers a unified theoretical framework to address the current crises in AI, but may also open a new era of next-generation AI driven by first principles and rigorously defined mathematics.

wchswchs 1y ago

History repeats itself — dynamic geometry is undergoing the same kind of development that the Turing machine once experienced: 1. The Turing machine was initially a theoretical success but physically impossible to build at the time. Dynamic geometry likewise has a self-consistent, rigorous mathematical theory, yet its numerical/computational costs are very large. 2. The Turing machine is a “finite-state machine” acting on an “infinite tape.” Dynamic geometry is an abstraction that uses “finite parameters” to represent an “infinite-dimensional” space. Both involve a move from the finite to the infinite. The Turing machine later became the theoretical foundation of the von Neumann architecture (the universal computer). Dynamic geometry needs an equivalent von Neumann — an architecture for AGI. This is the direction I am working toward: to realize a “von Neumann” moment for AI. When that moment arrives, the Turing Award will be virtually guaranteed. If the Turing machine made machines computable, then dynamic geometry makes intelligence computable. AGI would no longer be a qualitative or philosophical claim made by leading figures, it would be something that can be defined and computed mathematically.