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Fanghui Liu

Fanghui Liu contributes to research discovery and scholarly infrastructure.

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Machine Learning

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preprint2026arXiv

How does feature learning reshape the function space?

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.

preprint2026arXiv

Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

We study approximation by shallow ReLU$^s$ networks, $σ_s(t)=\max{0,t}^s$, and the generalization behavior of such networks under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,τ_d,s}$, $1\le p\le2$, we establish approximation bounds for shallow networks using spherical harmonic analysis. In particular, when the parameter measure is the uniform measure $τ_d$ and $p<p^*=(2d+2)/(d+3)$, we obtain the rate $O(m^{-1/2-d(2-p)/(2d(2-p)+2p(2s+d+1))}\log^{3/2}m)$, which improves the corresponding random-feature rate. We also derive approximation rates for Sobolev spaces $W^{α,p}$ in the range $1\le p<2$ by embedding them into spectral Barron spaces. Finally, for nonparametric regression with sub-Gaussian noise, we prove minimax-optimal generalization bounds for path-norm-regularized shallow ReLU$^s$ networks over Barron and Sobolev spaces, with matching lower bounds up to logarithmic factors.

preprint2021arXiv

Fast Learning in Reproducing Kernel Krein Spaces via Signed Measures

In this paper, we attempt to solve a long-lasting open question for non-positive definite (non-PD) kernels in machine learning community: can a given non-PD kernel be decomposed into the difference of two PD kernels (termed as positive decomposition)? We cast this question as a distribution view by introducing the \emph{signed measure}, which transforms positive decomposition to measure decomposition: a series of non-PD kernels can be associated with the linear combination of specific finite Borel measures. In this manner, our distribution-based framework provides a sufficient and necessary condition to answer this open question. Specifically, this solution is also computationally implementable in practice to scale non-PD kernels in large sample cases, which allows us to devise the first random features algorithm to obtain an unbiased estimator. Experimental results on several benchmark datasets verify the effectiveness of our algorithm over the existing methods.

preprint2021arXiv

Kernel regression in high dimensions: Refined analysis beyond double descent

In this paper, we provide a precise characterization of generalization properties of high dimensional kernel ridge regression across the under- and over-parameterized regimes, depending on whether the number of training data n exceeds the feature dimension d. By establishing a bias-variance decomposition of the expected excess risk, we show that, while the bias is (almost) independent of d and monotonically decreases with n, the variance depends on n, d and can be unimodal or monotonically decreasing under different regularization schemes. Our refined analysis goes beyond the double descent theory by showing that, depending on the data eigen-profile and the level of regularization, the kernel regression risk curve can be a double-descent-like, bell-shaped, or monotonic function of n. Experiments on synthetic and real data are conducted to support our theoretical findings.

preprint2020arXiv

Indefinite Kernel Logistic Regression with Concave-inexact-convex Procedure

In kernel methods, the kernels are often required to be positive definite, which restricts the use of many indefinite kernels. To consider those non-positive definite kernels, in this paper, we aim to build an indefinite kernel learning framework for kernel logistic regression. The proposed indefinite kernel logistic regression (IKLR) model is analysed in the Reproducing Kernel Kre\uın Spaces (RKKS) and then becomes non-convex. Using the positive decomposition of a non-positive definite kernel, the derived IKLR model can be decomposed into the difference of two convex functions. Accordingly, a concave-convex procedure is introduced to solve the non-convex optimization problem. Since the concave-convex procedure has to solve a sub-problem in each iteration, we propose a concave-inexact-convex procedure (CCICP) algorithm with an inexact solving scheme to accelerate the solving process. Besides, we propose a stochastic variant of CCICP to efficiently obtain a proximal solution, which achieves the similar purpose with the inexact solving scheme in CCICP. The convergence analyses of the above two variants of concave-convex procedure are conducted. By doing so, our method works effectively not only under a deterministic setting but also under a stochastic setting. Experimental results on several benchmarks suggest that the proposed IKLR model performs favorably against the standard (positive-definite) kernel logistic regression and other competitive indefinite learning based algorithms.

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