The Ladder of Emergence E: Information Cost Landscape and Gradient Flow — The Economic Geometry of Model Selection
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| author | zhou, changzheng zhou, ziqing |
| author_facet | zhou, changzheng zhou, ziqing |
| contents | <p>The core challenge of model selection lies in achieving an optimal balance be<br>tween descriptive complexity and predictive power. From the intersection of infor<br>mation theory and differential geometry, this paper constructs a rigorous mathe<br>matical framework of “cognitive economics” for this trade-off. The entire system<br>begins by recognizing the fundamental tension between classical Kolmogorov com<br>plexity and continuous geometric tools, bridging them via the minimum description<br>length principle, constructing the model space as the union of parametric statisti<br>cal manifolds. On this manifold, the descriptive complexity and predictive mutual<br>information of a model are unified into a smooth economic cost functional. This<br>paper rigorously proves that on this cost landscape, the natural gradient flow with<br>Fisher information as metric possesses local existence and uniqueness, and under<br>the assumptions of real analyticity of the cost and compactness of the physical sub<br>manifold, it converges to local minima via the Łojasiewicz inequality. Paradigm<br>emergence is interpreted as the gradient flow being obstructed at a low-dimensional<br>critical submanifold, then restarting through embedding into a higher-dimensional<br>Grassmannian tube. Subsequently, the three failure modes of hypothesis testing<br>are encoded into a sequential information filtration, and a rigorous channel separa<br>tion argument is given to prove that under a finite information budget, a three-layer<br>filtration is the minimal complete structure that can distinguish logical falsehood,<br>statistical unreliability, and insufficient precision. Finally, within the frameworks<br>of metric measure spaces and optimal transport, the pointwise gradient flow is lifted<br>to a Wasserstein gradient flow of probability measures, and the discrete paradigm<br>transition is quantified as the optimal stochastic control of a Schrödinger bridge<br>problem, whose minimum relative entropy gives the information cost and proba<br>bility of the transition. This framework provides a unified geometric and analytic<br>language for the cognitive economics of theory choice.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20019264 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The Ladder of Emergence E: Information Cost Landscape and Gradient Flow — The Economic Geometry of Model Selection zhou, changzheng zhou, ziqing model selection; cost landscape; information geometry; natural gradient f low; minimum description length; information filtration; optimal transport <p>The core challenge of model selection lies in achieving an optimal balance be<br>tween descriptive complexity and predictive power. From the intersection of infor<br>mation theory and differential geometry, this paper constructs a rigorous mathe<br>matical framework of “cognitive economics” for this trade-off. The entire system<br>begins by recognizing the fundamental tension between classical Kolmogorov com<br>plexity and continuous geometric tools, bridging them via the minimum description<br>length principle, constructing the model space as the union of parametric statisti<br>cal manifolds. On this manifold, the descriptive complexity and predictive mutual<br>information of a model are unified into a smooth economic cost functional. This<br>paper rigorously proves that on this cost landscape, the natural gradient flow with<br>Fisher information as metric possesses local existence and uniqueness, and under<br>the assumptions of real analyticity of the cost and compactness of the physical sub<br>manifold, it converges to local minima via the Łojasiewicz inequality. Paradigm<br>emergence is interpreted as the gradient flow being obstructed at a low-dimensional<br>critical submanifold, then restarting through embedding into a higher-dimensional<br>Grassmannian tube. Subsequently, the three failure modes of hypothesis testing<br>are encoded into a sequential information filtration, and a rigorous channel separa<br>tion argument is given to prove that under a finite information budget, a three-layer<br>filtration is the minimal complete structure that can distinguish logical falsehood,<br>statistical unreliability, and insufficient precision. Finally, within the frameworks<br>of metric measure spaces and optimal transport, the pointwise gradient flow is lifted<br>to a Wasserstein gradient flow of probability measures, and the discrete paradigm<br>transition is quantified as the optimal stochastic control of a Schrödinger bridge<br>problem, whose minimum relative entropy gives the information cost and proba<br>bility of the transition. This framework provides a unified geometric and analytic<br>language for the cognitive economics of theory choice.</p> |
| title | The Ladder of Emergence E: Information Cost Landscape and Gradient Flow — The Economic Geometry of Model Selection |
| topic | model selection; cost landscape; information geometry; natural gradient f low; minimum description length; information filtration; optimal transport |
| url | https://doi.org/10.5281/zenodo.20019264 |