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arxiv: 2605.17718 · v1 · pith:JC54ULOCnew · submitted 2026-05-18 · 📊 stat.ML · cs.LG

How does feature learning reshape the function space?

Jo\~ao Lobo , Bruno Loureiro , Long Tran-Than , Fanghui Liu This is my paper

Pith reviewed 2026-05-19 22:22 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords feature learningfunction spacegradient descenthigh-dimensional regimespiked covarianceadaptive kerneltwo-layer networksspectral structure
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The pith

In high dimensions, one large gradient step on a two-layer network produces features whose distribution approximates a target-dependent spiked Gaussian covariance, inducing a data-adaptive kernel that reshapes the function space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that feature learning during gradient descent is equivalent to a specific change in the distribution of network features rather than a simple rescaling of a fixed kernel. In the proportional high-dimensional limit, a sufficiently large update makes the post-training feature covariance look like a Gaussian with an extra spike aligned to the target function. This change creates an adaptive kernel whose eigenstructure favors directions that match the signal in the data. A reader should care because it supplies a concrete function-space account of why neural networks can represent different functions after training compared with static kernel methods.

Core claim

We prove that, in the high-dimensional proportional regime, after a large gradient step the post-update feature distribution is well approximated by a target-dependent spiked Gaussian covariance. This induces a data-adaptive kernel that reshapes the function space and modifies its spectral structure. Feature learning can be viewed as a distributional transformation in parameter space or input space, or equivalently as the introduction of a target-dependent kernel. In particular, the update selectively amplifies eigenvalues aligned with the target direction and mixes leading eigenfunctions, coupling the top radial mode with a target-aligned quadratic harmonic. The overall effect is a data-adp

What carries the argument

Target-dependent spiked Gaussian covariance that approximates the post-update feature distribution and thereby induces the data-adaptive kernel.

If this is right

  • The induced kernel selectively amplifies eigenvalues aligned with the target direction.
  • Leading eigenfunctions mix, coupling the top radial mode with a target-aligned quadratic harmonic.
  • Feature learning acts as a distributional transformation in parameter space or input space.
  • Early training deforms the function space to preferentially enhance directions aligned with the signal rather than rescaling a fixed kernel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Repeated gradient steps could compound the distributional shift, producing successively more adapted kernels at later training stages.
  • The same spiked approximation may appear in deeper networks if each layer experiences an analogous large update.
  • Low-dimensional regimes or small step sizes offer a direct test of where the reshaping mechanism breaks down.
  • This view connects to analyses of kernel evolution under gradient flow by supplying an explicit distributional mechanism for the adaptation.

Load-bearing premise

The analysis requires the high-dimensional proportional regime together with a large enough gradient step size so that the updated features can be approximated by the spiked Gaussian form.

What would settle it

Numerical computation of the empirical covariance of hidden features after one large gradient step on finite but proportional n and d data, checking whether the observed matrix deviates from the predicted target-dependent spike.

Figures

Figures reproduced from arXiv: 2605.17718 by Bruno Loureiro, Fanghui Liu, Jo\~ao Lobo, Long Tran-Than.

Figure 1
Figure 1. Figure 1: (a) Alignment with the directional feature [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

Feature learning is widely regarded as the key mechanism distinguishing neural networks from fixed-kernel methods, yet its impact on the induced function space remains poorly understood. In this work, we precisely characterize how the function space spanned by the features of a two-layer neural network evolves during gradient descent training. We prove that, in the high-dimensional proportional regime, after a large gradient step the post-update feature distribution is well approximated by a target-dependent spiked Gaussian covariance. This induces a data-adaptive kernel that reshapes the function space and modifies its spectral structure. Our analysis reveals that feature learning can be interpreted as a distributional transformation in either parameter space or input space, equivalently as the introduction of a target-dependent kernel. In particular, it selectively amplifies eigenvalues aligned with the target direction and mixes leading eigenfunctions, coupling the top radial mode with a target-aligned quadratic harmonic. Overall, our results provide a precise function-space perspective on early-stage feature learning: rather than just rescaling a fixed kernel, gradient descent induces a data-adaptive deformation that preferentially enhances directions aligned with the signal in the data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that in the high-dimensional proportional regime (n, d → ∞ with n/d fixed), a single large gradient step on a two-layer neural network produces a post-update feature distribution that is well approximated by a target-dependent spiked Gaussian covariance. This induces a data-adaptive kernel that reshapes the function space, selectively amplifying eigenvalues aligned with the target direction and mixing leading eigenfunctions (e.g., coupling the top radial mode with a target-aligned quadratic harmonic). Feature learning is interpreted equivalently as a distributional transformation in parameter space or input space.

Significance. If the central approximation holds with the stated precision, the work supplies a rigorous function-space view of early-stage feature learning that goes beyond fixed-kernel or NTK analyses by exhibiting an explicit data-adaptive deformation of the spectral structure. The result is potentially useful for understanding how gradient descent modifies the effective kernel during the initial phase of training and for designing adaptive kernels that capture target-aligned directions.

major comments (2)
  1. [§3.1, Theorem 1] §3.1, Theorem 1: The spiked-Gaussian approximation is asserted to hold after a 'sufficiently large' gradient step, yet the statement provides neither an explicit lower bound on the step size η nor quantitative error bounds (in total variation or Wasserstein distance) that depend on n, d, and η. Without these, it is impossible to verify the domain of validity of the claimed limit or to assess how the approximation degrades when the step-size condition is relaxed.
  2. [§4.2, Eq. (17)] §4.2, Eq. (17): The induced kernel is defined via the expectation over the spiked covariance; however, the derivation of the eigenvalue amplification and eigenfunction mixing (radial mode coupled to quadratic harmonic) appears to rest on an additional assumption that the target is exactly aligned with a single direction. The paper does not state whether this alignment is necessary or how the spectral reshaping generalizes to targets with multiple relevant directions.
minor comments (2)
  1. [§2] Notation for the proportional limit (n/d → γ) is introduced in §2 but used inconsistently in the statement of the main result; a single displayed definition would improve readability.
  2. [Figure 2] Figure 2 caption does not specify the value of the step size η used in the simulation, making it difficult to relate the plotted spectra to the theoretical regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope and presentation of our results. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3.1, Theorem 1] §3.1, Theorem 1: The spiked-Gaussian approximation is asserted to hold after a 'sufficiently large' gradient step, yet the statement provides neither an explicit lower bound on the step size η nor quantitative error bounds (in total variation or Wasserstein distance) that depend on n, d, and η. Without these, it is impossible to verify the domain of validity of the claimed limit or to assess how the approximation degrades when the step-size condition is relaxed.

    Authors: We agree that greater precision on the step-size condition would improve the statement. In the proof of Theorem 1 the requirement that η be sufficiently large arises from ensuring the target-dependent spike dominates the fluctuation terms in the high-dimensional limit; this translates to η exceeding a constant determined by the data variance and the Lipschitz constant of the activation. We will revise the theorem to state an explicit lower bound of this form. Our analysis is strictly asymptotic (n,d→∞ with n/d fixed), so we do not derive non-asymptotic total-variation or Wasserstein bounds; we will add a remark noting this limitation and the resulting domain of validity. revision: partial

  2. Referee: [§4.2, Eq. (17)] §4.2, Eq. (17): The induced kernel is defined via the expectation over the spiked covariance; however, the derivation of the eigenvalue amplification and eigenfunction mixing (radial mode coupled to quadratic harmonic) appears to rest on an additional assumption that the target is exactly aligned with a single direction. The paper does not state whether this alignment is necessary or how the spectral reshaping generalizes to targets with multiple relevant directions.

    Authors: The single-direction setting is adopted for expository clarity, as it already exhibits the essential phenomenon of target-dependent eigenvalue amplification and the specific radial-to-quadratic mixing. The underlying spiked-covariance construction extends immediately to a finite number of spikes aligned with a multi-dimensional target subspace; the induced kernel then amplifies the corresponding eigenspace and produces analogous mixing within that subspace. We will revise §4.2 to state the single-direction assumption explicitly and add a short paragraph describing the multi-spike generalization, confirming that the qualitative conclusions on data-adaptive reshaping remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit high-dimensional limit assumptions without reduction to fitted inputs or self-referential definitions

full rationale

The paper derives the spiked Gaussian covariance approximation for post-update features from gradient descent dynamics under the stated proportional regime (n,d→∞, n/d fixed) and large step-size condition. This is presented as a proven limit result rather than an ansatz, fit, or self-definition. No equations reduce the target-dependent kernel or spectral reshaping to a tautology or to a parameter fitted from the same data being predicted. Self-citations, if present, are not load-bearing for the central claim, which rests on the regime-specific analysis rather than prior author work invoked as uniqueness. The result is not a renaming of a known pattern but a characterization of function-space evolution. The derivation chain is therefore self-contained against the explicit assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the high-dimensional proportional limit and the large-gradient-step approximation; these are standard modeling choices in the field but constitute the main unverified assumptions for the result.

axioms (2)
  • domain assumption High-dimensional proportional regime: n, d → ∞ with n/d = γ fixed
    Invoked to obtain the spiked Gaussian approximation after one gradient step
  • domain assumption Sufficiently large gradient step size
    Required for the post-update feature distribution to concentrate around the target-dependent spike

pith-pipeline@v0.9.0 · 5722 in / 1441 out tokens · 33307 ms · 2026-05-19T22:22:46.075133+00:00 · methodology

discussion (0)

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Reference graph

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