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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.26266 |
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| _version_ | 1866917229715521536 |
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| author | Rygaard, Frederik Möbius Zhu, Shen Jin, Yinzhu Hauberg, Søren Fletcher, Tom |
| author_facet | Rygaard, Frederik Möbius Zhu, Shen Jin, Yinzhu Hauberg, Søren Fletcher, Tom |
| contents | The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Likely Geometry of Generative Models Rygaard, Frederik Möbius Zhu, Shen Jin, Yinzhu Hauberg, Søren Fletcher, Tom Machine Learning The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets. |
| title | A Likely Geometry of Generative Models |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.26266 |