TL;DR: Natural images live on a hypersphere — and treating them that way improves flow matching.<br>Geometry-aware generative modeling has worked well on known manifolds (molecules, crystals, proteins), but natural images have stayed stuck in Euclidean space because nobody knew what manifold they lived on. </p>\n<p>We show a surprisingly simple answer: their semantic content is almost entirely in the direction, not the norm. Projecting images (both RGB and VAE latents) onto a sphere of the dataset's mean radius leaves them perceptually indistinguishable from the originals.</p>\n<p>Building on this, we propose SOT-CFM (angular OT cost) and SFM (fully Riemannian flow matching on the sphere). SFM is, to our knowledge, the first successful application of a fully manifold-based generative framework to large-scale natural images.</p>\n","updatedAt":"2026-05-26T02:11:50.938Z","author":{"_id":"64a7c64d01646254507a644c","avatarUrl":"/avatars/79ae2bfc5eb9eaeca53e5d2c9cc10831.svg","fullname":"junho lee","name":"isno0907","type":"user","isPro":false,"isHf":false,"isHfAdmin":false,"isMod":false,"followerCount":1,"isUserFollowing":false}},"numEdits":0,"identifiedLanguage":{"language":"en","probability":0.921486496925354},"editors":["isno0907"],"editorAvatarUrls":["/avatars/79ae2bfc5eb9eaeca53e5d2c9cc10831.svg"],"reactions":[],"isReport":false}}],"primaryEmailConfirmed":false,"paper":{"id":"2605.25294","authors":[{"_id":"6a14ffafb57a1823d5708a16","name":"Junho Lee","hidden":false},{"_id":"6a14ffafb57a1823d5708a17","name":"Kwanseok Kim","hidden":false},{"_id":"6a14ffafb57a1823d5708a18","name":"Joonseok Lee","hidden":false}],"publishedAt":"2026-05-24T00:00:00.000Z","submittedOnDailyAt":"2026-05-26T00:00:00.000Z","title":"Geometry-Aware Image Flow Matching","submittedOnDailyBy":{"_id":"64a7c64d01646254507a644c","avatarUrl":"/avatars/79ae2bfc5eb9eaeca53e5d2c9cc10831.svg","isPro":false,"fullname":"junho lee","user":"isno0907","type":"user","name":"isno0907"},"summary":"Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.","upvotes":7,"discussionId":"6a14ffb0b57a1823d5708a19","ai_summary":"Geometry-aware generative models leveraging spherical manifolds and optimal transport techniques outperform traditional Euclidean approaches for natural image synthesis.","ai_keywords":["generative models","geometry-aware modeling","manifold-constrained settings","natural images","hypersphere","Spherical Optimal Transport Flow Matching","Spherical Flow Matching","Riemannian manifold-based modeling"]},"canReadDatabase":false,"canManagePapers":false,"canSubmit":false,"hasHfLevelAccess":false,"upvoted":false,"upvoters":[{"_id":"64a7c64d01646254507a644c","avatarUrl":"/avatars/79ae2bfc5eb9eaeca53e5d2c9cc10831.svg","isPro":false,"fullname":"junho lee","user":"isno0907","type":"user"},{"_id":"68d66a2bbfd2620af98bec48","avatarUrl":"/avatars/41e394fe47764b343422d4418270912b.svg","isPro":false,"fullname":"Zuowu Shi","user":"RL4LLM4AI","type":"user"},{"_id":"68d66b1178d69b134522ae80","avatarUrl":"/avatars/63066f32be1142fdd09bcfea1ea6e823.svg","isPro":false,"fullname":"Baizhou Zhang","user":"Fridge003","type":"user"},{"_id":"6902f4ecd642fc67f25f41d4","avatarUrl":"/avatars/c5237e662ac5e83b7a2ca2dacf95c49d.svg","isPro":false,"fullname":"Kwanseok Kim","user":"Kwanseok-k","type":"user"},{"_id":"6559a64c727df37c7772d06f","avatarUrl":"/avatars/7e33d76812209daf708a328f552f86ca.svg","isPro":false,"fullname":"Junhyeong Park","user":"junbro1016","type":"user"},{"_id":"69d1d23a3ce8c7cbe755616b","avatarUrl":"/avatars/bba84639fc9c6456b0f387bbcc7d1a37.svg","isPro":false,"fullname":"faye","user":"fayej1221","type":"user"},{"_id":"65f4082041bafd040f7c6c04","avatarUrl":"/avatars/779a7ae2f4e81a104e96521cc3c4c64d.svg","isPro":false,"fullname":"DevLee","user":"DevLee","type":"user"}],"acceptLanguages":["en"],"dailyPaperRank":0,"markdownContentUrl":"https://huggingface.co/buckets/huggingchat/papers-content/resolve/2605/2605.25294.md"}">
Geometry-Aware Image Flow Matching
Abstract
Geometry-aware generative models leveraging spherical manifolds and optimal transport techniques outperform traditional Euclidean approaches for natural image synthesis.
AI-generated summary
Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.
Community
TL;DR: Natural images live on a hypersphere — and treating them that way improves flow matching.
Geometry-aware generative modeling has worked well on known manifolds (molecules, crystals, proteins), but natural images have stayed stuck in Euclidean space because nobody knew what manifold they lived on.
We show a surprisingly simple answer: their semantic content is almost entirely in the direction, not the norm. Projecting images (both RGB and VAE latents) onto a sphere of the dataset's mean radius leaves them perceptually indistinguishable from the originals.
Building on this, we propose SOT-CFM (angular OT cost) and SFM (fully Riemannian flow matching on the sphere). SFM is, to our knowledge, the first successful application of a fully manifold-based generative framework to large-scale natural images.
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Cite arxiv.org/abs/2605.25294 in a model README.md to link it from this page.
Cite arxiv.org/abs/2605.25294 in a dataset README.md to link it from this page.
Cite arxiv.org/abs/2605.25294 in a Space README.md to link it from this page.
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