Shallow ReLU$^s$ Networks in $L^p$-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization
arXiv:2605.18468v2 Announce Type: replace-cross Abstract: We study approximation by shallow ReLU$^s$ networks, $\sigma_s(t)=\max{0,t}^s$, and the generalization behavior of such networks under $\ell_1$ path-norm control. For the $L^p$-type integral spaces $\widetilde{\mathcal{F}}_{p,\tau_d,s}$, $1\le p\le2$, we establish approximation bounds for shallow networks using spherical harmonic analysis. In particular, when the parameter measure is the uniform measure $\tau_d$ 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 correspo
This is a new academic publication (arXiv v2) in the field of machine learning theory, building on existing research.
This paper offers incremental theoretical advancements in neural network approximation and generalization, which are foundational but not immediately actionable for strategic readers.
This research refines the mathematical understanding of certain ReLU network architectures and their performance guarantees, but it does not introduce a new paradigm or practical breakthrough.
Further theoretical development in machine learning interpretability and performance bounds.
Potential for slightly more optimized or robust small-scale neural network models in academic or highly specialized applications over a very long time horizon.
Extremely long-term, this type of foundational work might contribute to the design principles for more efficient AI hardware or algorithms.
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