{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/compressing-invariant-manifolds-in-neural","title":"Geometric compression of invariant manifolds in neural nets","arxiv_id":"2007.11471","date":"2020-07-22","proceeding":null,"authors":["Jonas Paccolat","Leonardo Petrini","Mario Geiger","Kevin Tyloo","Matthieu Wyart"],"abstract":"We study how neural networks compress uninformative input space in models where data lie in $d$ dimensions, but whose label only vary within a linear manifold of dimension $d_\\parallel < d$. We show that for a one-hidden layer network initialized with infinitesimal weights (i.e. in the feature learning regime) trained with gradient descent, the first layer of weights evolve to become nearly insensitive to the $d_\\perp=d-d_\\parallel$ uninformative directions. These are effectively compressed by a factor $\\lambda\\sim \\sqrt{p}$, where $p$ is the size of the training set. We quantify the benefit of such a compression on the test error $\\epsilon$. For large initialization of the weights (the lazy training regime), no compression occurs and for regular boundaries separating labels we find that $\\epsilon \\sim p^{-\\beta}$, with $\\beta_\\text{Lazy} = d / (3d-2)$. Compression improves the learning curves so that $\\beta_\\text{Feature} = (2d-1)/(3d-2)$ if $d_\\parallel = 1$ and $\\beta_\\text{Feature} = (d + d_\\perp/2)/(3d-2)$ if $d_\\parallel > 1$. We test these predictions for a stripe model where boundaries are parallel interfaces ($d_\\parallel=1$) as well as for a cylindrical boundary ($d_\\parallel=2$). Next we show that compression shapes the Neural Tangent Kernel (NTK) evolution in time, so that its top eigenvectors become more informative and display a larger projection on the labels. Consequently, kernel learning with the frozen NTK at the end of training outperforms the initial NTK. We confirm these predictions both for a one-hidden layer FC network trained on the stripe model and for a 16-layers CNN trained on MNIST, for which we also find $\\beta_\\text{Feature}>\\beta_\\text{Lazy}$.","url_abs":"https://arxiv.org/abs/2007.11471v4","url_pdf":"https://arxiv.org/pdf/2007.11471v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"compressing-invariant-manifolds-in-neural","repo_url":"https://github.com/mariogeiger/feature_lazy","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[],"methods":[{"method_slug":"ntk","method_name":"NTK"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2007.11471","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2007.11471"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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