Average Gradient Outer Product in kernel regression provably recovers the central subspace for multi-index models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916013344292864 |
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| author | Zhu, Libin Davis, Damek Drusvyatskiy, Dmitriy Fazel, Maryam |
| author_facet | Zhu, Libin Davis, Damek Drusvyatskiy, Dmitriy Fazel, Maryam |
| contents | 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 Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions 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 $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15082 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Average Gradient Outer Product in kernel regression provably recovers the central subspace for multi-index models Zhu, Libin Davis, Damek Drusvyatskiy, Dmitriy Fazel, Maryam Machine Learning Statistics Theory 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 Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions 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 $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice. |
| title | Average Gradient Outer Product in kernel regression provably recovers the central subspace for multi-index models |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2605.15082 |