Demystify how neural network learn group composition from a representation-theoretical perspective.</p>\n","updatedAt":"2026-06-04T19:03:42.781Z","author":{"_id":"6852f69c7b834665dc38c837","avatarUrl":"/avatars/bf9c5fc72300756e88319ec45c75f337.svg","fullname":"Jianliang He","name":"JLiangHe","type":"user","isPro":false,"isHf":false,"isHfAdmin":false,"isMod":false,"followerCount":5,"isUserFollowing":false}},"numEdits":0,"identifiedLanguage":{"language":"en","probability":0.8372782468795776},"editors":["JLiangHe"],"editorAvatarUrls":["/avatars/bf9c5fc72300756e88319ec45c75f337.svg"],"reactions":[],"isReport":false}}],"primaryEmailConfirmed":false,"paper":{"id":"2606.02993","authors":[{"_id":"6a202b6015100c5272a8418b","name":"Jianliang He","hidden":false},{"_id":"6a202b6015100c5272a8418c","name":"Leda Wang","hidden":false},{"_id":"6a202b6015100c5272a8418d","name":"Fengzhuo Zhang","hidden":false},{"_id":"6a202b6015100c5272a8418e","name":"Siyu Chen","hidden":false},{"_id":"6a202b6015100c5272a8418f","name":"Zhuoran Yang","hidden":false}],"publishedAt":"2026-06-02T00:00:00.000Z","submittedOnDailyAt":"2026-06-04T00:00:00.000Z","title":"Neural Networks Provably Learn Spectral Representations for Group Composition","submittedOnDailyBy":{"_id":"6852f69c7b834665dc38c837","avatarUrl":"/avatars/bf9c5fc72300756e88319ec45c75f337.svg","isPro":false,"fullname":"Jianliang He","user":"JLiangHe","type":"user","name":"JLiangHe"},"summary":"Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict g_1 star g_2 for elements of a finite group G. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.","upvotes":4,"discussionId":"6a202b6015100c5272a84190","ai_summary":"Neural network training on group composition tasks exhibits convergence to irreducible representations and rotational rank-one alignment through Riemannian gradient ascent on representation-theoretic energy functionals.","ai_keywords":["group composition task","two-layer neural network","finite group","projected gradient flow","Fourier domain","Riemannian gradient ascent","representation-theoretic energy functional","irreducible representation","cross-layer Fourier coefficients","rotational rank-one alignment","matrix-valued group representations","Haar-uniform phases","majority-vote mechanism","exponential convergence rates"],"ai_summary_model":"Qwen/Qwen2.5-Coder-32B-Instruct"},"canReadDatabase":false,"canManagePapers":false,"canSubmit":false,"hasHfLevelAccess":false,"upvoted":false,"upvoters":[{"_id":"6852f69c7b834665dc38c837","avatarUrl":"/avatars/bf9c5fc72300756e88319ec45c75f337.svg","isPro":false,"fullname":"Jianliang He","user":"JLiangHe","type":"user"},{"_id":"64b8c1a995bd42c7707f7918","avatarUrl":"/avatars/08c2929f8f150ecd6f8e5a06c4cb9034.svg","isPro":true,"fullname":"Fengzhuo Zhang","user":"Fengzhuo","type":"user"},{"_id":"62ea79dd01ed9b0e8f61ccd3","avatarUrl":"/avatars/70af83e0e267be39fcd5f23b85e2dafa.svg","isPro":false,"fullname":"Chengsong Huang","user":"ChengsongHuang","type":"user"},{"_id":"6a21dbac399352692f5d4154","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/6a21dbac399352692f5d4154/HkeXK_Rex79fYfvgQm5yH.jpeg","isPro":false,"fullname":"Perma Frost","user":"LqJia","type":"user"}],"acceptLanguages":["en"],"dailyPaperRank":0}">

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