{"paper":{"title":"Average Gradient Outer Product in kernel regression provably recovers the central subspace for multi-index models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The average gradient outer product from kernel ridge regression recovers the central subspace of multi-index models in sample regimes too small for accurate prediction.","cross_cats":["cs.LG","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Damek Davis, Dmitriy Drusvyatskiy, Libin Zhu, Maryam Fazel","submitted_at":"2026-05-14T17:05:30Z","abstract_excerpt":"We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\\in\\mathbb{R}^{r\\times d}$ and $r\\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the top r-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function f* has degree p*, ... subspace recovery occurs in the much lower sample regime n ≍ d^{p+δ} for any δ ∈ (0,1).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"a low degree p component of f* already carries all relevant directions for prediction, together with unspecified 'reasonable assumptions' on the kernel and the multi-index structure (abstract only).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For multi-index polynomials, the top r eigenspace of the AGOP matrix from KRR recovers the central subspace at sample complexity n ~ d^{p+δ} where p is the degree of the informative component.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The average gradient outer product from kernel ridge regression recovers the central subspace of multi-index models in sample regimes too small for accurate prediction.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"45310b90d8bc0c84b1d0a09e96c47bebf99c9878d66db731277a0f6de1693b7e"},"source":{"id":"2605.15082","kind":"arxiv","version":1},"verdict":{"id":"a08aba0c-35c9-45ae-baca-7e51e7df039e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:05:53.439476Z","strongest_claim":"the top r-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function f* has degree p*, ... subspace recovery occurs in the much lower sample regime n ≍ d^{p+δ} for any δ ∈ (0,1).","one_line_summary":"For multi-index polynomials, the top r eigenspace of the AGOP matrix from KRR recovers the central subspace at sample complexity n ~ d^{p+δ} where p is the degree of the informative component.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"a low degree p component of f* already carries all relevant directions for prediction, together with unspecified 'reasonable assumptions' on the kernel and the multi-index structure (abstract only).","pith_extraction_headline":"The average gradient outer product from kernel ridge regression recovers the central subspace of multi-index models in sample regimes too small for accurate prediction."},"references":{"count":64,"sample":[{"doi":"","year":2023,"title":"Sgd learning on neural networks: leap complexity and saddle-to-saddle dynamics","work_id":"c710cb1d-e477-4ec1-91b5-a8534000fc4e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks","work_id":"4d8fa6c0-826f-4638-986c-6f3c0d96af37","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Classical orthogonal polynomials","work_id":"93ead7bf-64f0-4d79-9e77-66dbc737e138","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Springer. 2006, pp. 36–62","work_id":"151a1b61-ae68-4f67-b38b-57c5462d3dea","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Online learning and information exponents: On the importance of batch size, and time/complexity tradeoffs","work_id":"0e969c0f-81e0-4313-af54-714970b4945a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":64,"snapshot_sha256":"d40a3fedb25ffe5dd07b0cccca202bc662cdbfc53b810f036b02bd0af43a4d99","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"be6fd5681d87fd153a1276402e7e512cbf7be9cc4564aa2c216aaf765ed23e22"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}