New theory explores reproducing kernel spaces with $\ell^p$ norms

This paper introduces a theoretical framework for reproducing kernel quasi-Banach spaces (RKQBSs) using $\ell^p$ quasi-norms, particularly focusing on the non-convex regime where $0<p<1$. The research extends previous work on $\ell^1$ regularization to achieve stronger sparsity, akin to the $\ell^0$ pseudo-norm. The study establishes existence and sparsity results for interpolation and regularized learning, characterizes representer theorems using a generalized Lebesgue constant, and derives bounds under various assumptions. For $p>1$, the paper constructs RKBSs with a fixed dictionary and identifies an optimal relaxed representer factor. Numerical experiments are conducted to compare the accuracy and sparsity across the different $p$ regimes. AI

IMPACT Advances theoretical understanding of kernel methods and sparsity in machine learning.

RANK_REASON This is a theoretical computer science paper published on arXiv, detailing mathematical advancements in machine learning theory. [lever_c_demoted from research: ic=1 ai=1.0]

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• arXivLabs
• CORE Recommender
• $\ell^0$ pseudo-norm
• $\ell^1$ norm
• $\ell^p$ Coefficient Norms
• $\ell^p$ quasi-norms
• IArxiv Recommender
• Lebesgue constant
• Reproducing Kernel Banach Spaces
• Reproducing Kernel Quasi-Banach Spaces
• RKBSs
• RKQBSs

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