Approximation by Neural Network Sampling Operators in Mixed Lebesgue Spaces

TL;DR

The paper addresses the problem of quantifying the rate of approximation for Neural Network Sampling Operators in mixed Lebesgue spaces $L^{p,q}$ using the $L^{p,q}$-averaged modulus of smoothness $\tau_r$. It introduces the subspaces $\Lambda^{p,q}$ via admissible partitions to ensure well-defined sampling operators and finiteness of the modulus, and develops an interpolation framework based on Steklov-type approximants. A Jackson-type result is established: neural-network operators converge at a rate governed by $\tau_r$, yielding explicit bounds $\|F_n f-f\|_{p,q} \le C_1\tau_1\left(f;1/n\right)_{p,q} + C_2\tau_2\left(f;1/n\right)_{p,q}$. The work is complemented by concrete examples with sigmoidal activations (e.g., logistic, tanh), showing that continuous and discontinuous targets can be implemented and approximated within the mixed-norm setting, thereby linking theoretical rates to practical NN-based sampling tasks.

Abstract

In this paper, we prove the rate of approximation for the Neural Network Sampling Operators activated by sigmoidal functions with mixed Lebesgue norm in terms of averaged modulus of smoothness for a bounded measurable functions on bounded domain. In order to achieve the above result, we first establish that the averaged modulus of smoothness is finite for certain suitable subspaces of $L^{p,q}(\mathbb{R}\times\mathbb{R}).$ Using the properties of averaged modulus of smoothness, we estimate the rate of approximation of certain linear operators in mixed Lebesgue norm. Then, as an application of these linear operators, we obtain the Jackson type approximation theorem, in order to give a characterization for the rate of approximation of neural network operators in-terms of averaged modulus of smoothness in mixed norm. Lastly, we discuss some examples of sigmoidal functions and using these sigmoidal functions, we show the implementation of continuous and discontinuous functions by neural network operators.

Figures (6)

• Figure 1: The function $f$
• Figure 2: Approximation $F_{20}f$ of $f$ using sigmoidal function $\sigma_{l}$.
• Figure 3: Approximation $F_{20}f$ of $f$ using hyperbolic tangent function .
• Figure 4: The function $g$
• Figure 5: Approximation $F_{20}g$ of $g$ using sigmoidal function.

Theorems & Definitions (54)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Remark 1
• Definition 6
• Proposition 1
• proof
• Lemma 1