{"paper":{"title":"Scaling Laws from Sequential Feature Recovery: A Solvable Hierarchical Model","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A layer-wise spectral algorithm on a hierarchical target with power-law feature weights recovers latent directions sequentially and aggregates their sharp thresholds into an explicit power-law decay of prediction error.","cross_cats":["cs.LG","math.PR","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Arie Wortsman-Zurich, Bruno Loureiro, Florent Krzakala, Hugo Tabanelli, Yatin Dandi","submitted_at":"2026-05-14T08:37:28Z","abstract_excerpt":"We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination of latent compositional features whose weights decrease as a power law. We show that a layer-wise spectral algorithm adapted to this compositional structure achieves improved scaling relative to shallow, non-adaptive methods, and recovers the latent directions sequentially: strong features become detectable at small sample sizes, while weaker features requir"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"aggregating these transitions yields an explicit power-law decay of the prediction error","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"the high-dimensional target admits a representation as a combination of latent compositional features whose weights decrease as a power law, and that the layer-wise spectral algorithm is specifically adapted to this compositional structure","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A solvable hierarchical model with power-law feature strengths yields explicit power-law scaling of prediction error through sequential recovery of latent directions by a layer-wise spectral algorithm.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A layer-wise spectral algorithm on a hierarchical target with power-law feature weights recovers latent directions sequentially and aggregates their sharp thresholds into an explicit power-law decay of prediction error.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f80eb6c129438bd9d298e5e3fd1390cce232e3daf334947995d4fcfcfa3afbc3"},"source":{"id":"2605.14567","kind":"arxiv","version":1},"verdict":{"id":"6e1da657-0718-4853-a364-30dbde508cf1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:35:40.345991Z","strongest_claim":"aggregating these transitions yields an explicit power-law decay of the prediction error","one_line_summary":"A solvable hierarchical model with power-law feature strengths yields explicit power-law scaling of prediction error through sequential recovery of latent directions by a layer-wise spectral algorithm.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"the high-dimensional target admits a representation as a combination of latent compositional features whose weights decrease as a power law, and that the layer-wise spectral algorithm is specifically adapted to this compositional structure","pith_extraction_headline":"A layer-wise spectral algorithm on a hierarchical target with power-law feature weights recovers latent directions sequentially and aggregates their sharp thresholds into an explicit power-law decay of prediction error."},"references":{"count":187,"sample":[{"doi":"","year":2022,"title":"Random matrix methods for machine learning , author=. 2022 , publisher=","work_id":"7cb35935-4ffa-43b3-b4ff-744e845a45b5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Introduction to the non-asymptotic analysis of random matrices","work_id":"695fe191-d99a-4805-9596-420902e88b7f","ref_index":2,"cited_arxiv_id":"1011.3027","is_internal_anchor":true},{"doi":"","year":2022,"title":"Applied and Computational Harmonic Analysis , volume=","work_id":"e917bb1e-a824-4914-a667-682624bbbf0b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Bert: Pre-training of deep bidirectional transformers for language understanding , author=. 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