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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2508.13658 |
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| _version_ | 1866909743143976960 |
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| author | Alpay, Faruk Alakkad, Hamdi |
| author_facet | Alpay, Faruk Alakkad, Hamdi |
| contents | We develop a calibrated diffusion framework by synthesizing three established concepts: linear Laplacian smoothing, nonlinear graph p-Laplacian flows, and a learnable dissipation term derived from a strongly convex potential. This synthesis provides a general model for graph-based diffusion with controllable dynamics. Our key theoretical results include a quantified two-regime decay analysis for $p>2$, which provides stronger, p-dependent transient bounds not captured by standard ISS templates, and the first formalization of a "non-synonymy" impossibility principle, which proves that fixed-parameter models cannot meet universal performance targets across graphs with varying spectral properties. To address this, we propose a constructive calibration algorithm (SGPS) with formal guarantees for achieving target rates and mass. We derive explicit, closed-form lower bounds for the graph p-gap on canonical graphs a notable improvement over prior implicit estimates and provide sharp constants for discrete-time and stochastic stability, including a contextualized restatement of the necessary and sufficient Euler step-size and a strengthened analysis of the stochastic noise floor. Illustrative, small-scale empirical validations confirm the tightness of key theoretical bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Calibrated Semantic Diffusion: A p-Laplacian Synthesis with Learnable Dissipation, Quantified Constants, and Graph-Aware Calibration Alpay, Faruk Alakkad, Hamdi Optimization and Control Analysis of PDEs Dynamical Systems 68T07, 35K55, 93D20, 35B40, 05C50 We develop a calibrated diffusion framework by synthesizing three established concepts: linear Laplacian smoothing, nonlinear graph p-Laplacian flows, and a learnable dissipation term derived from a strongly convex potential. This synthesis provides a general model for graph-based diffusion with controllable dynamics. Our key theoretical results include a quantified two-regime decay analysis for $p>2$, which provides stronger, p-dependent transient bounds not captured by standard ISS templates, and the first formalization of a "non-synonymy" impossibility principle, which proves that fixed-parameter models cannot meet universal performance targets across graphs with varying spectral properties. To address this, we propose a constructive calibration algorithm (SGPS) with formal guarantees for achieving target rates and mass. We derive explicit, closed-form lower bounds for the graph p-gap on canonical graphs a notable improvement over prior implicit estimates and provide sharp constants for discrete-time and stochastic stability, including a contextualized restatement of the necessary and sufficient Euler step-size and a strengthened analysis of the stochastic noise floor. Illustrative, small-scale empirical validations confirm the tightness of key theoretical bounds. |
| title | Calibrated Semantic Diffusion: A p-Laplacian Synthesis with Learnable Dissipation, Quantified Constants, and Graph-Aware Calibration |
| topic | Optimization and Control Analysis of PDEs Dynamical Systems 68T07, 35K55, 93D20, 35B40, 05C50 |
| url | https://arxiv.org/abs/2508.13658 |