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Life sciences · Preprint
arXiv · September 4, 2026
Raises a question worth testing. It does not answer one.
This is a theoretical mathematics paper deriving approximation rates for shallow neural networks with general activation functions in mixed Sobolev spaces. The work establishes a Fourier-block principle and identifies optimal algebraic approximation exponents for specific activations (ReLU^k, B-splines, ELU, cosine), but does not report empirical validation or application to clinical or biological systems.
Preprint.
Global approximation rate has algebraic order min{α,ρ} for target functions of mixed smoothness α, where ρ is univariate approximation order For ReLU^k, optimal algebraic approximation exponent is min{α,k+1} up to logarithmic factors Exponent min{α,k+1} achieved for cardinal B-splines and soft-ReLU^k
Safety was not reported in the material analysed. Check the source before drawing any conclusion about harm.
The source did not state who this applies to in practice.
This is a theoretical mathematical analysis establishing approximation bounds for neural networks; it addresses a mechanistic question about activation function properties rather than empirical clinical or biological evidence.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
What is missing. This record has no reported figures. That is a gap in the analysis, not a judgement about the study.
We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{α,ρ\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{α,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{α,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.
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