On best approximation by multivariate ridge functions with applications to generalized translation networks

We prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n h_k(A_k x) \in \mathbb{R}$ with $h_k : \mathbb{R}^\ell \to \mathbb{R}$ and $A_k \in \mathbb{R}^{\ell \times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell)}$, where $r$ is the regularity of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty$-Sobolev functions of regularity $r$ with error measured in $L^1$, while our upper bound applies to the approximation of $L^p$-Sobolev functions in $L^p$ for any $1 \leq p \leq \infty$. These bounds generalize well-known results about the approximation properties of univariate ridge functions to the multivariate case. Moreover, we use these bounds to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.

Efficient uniform approximation using Random Vector Functional Link networks

A Random Vector Functional Link (RVFL) network is a depth-2 neural network with random inner weights and biases. As only the outer weights of such architectures need to be learned, the learning process boils down to a linear optimization task, allowing one to sidestep the pitfalls of nonconvex optimization problems. In this paper, we prove that an RVFL with ReLU activation functions can approximate Lipschitz continuous functions provided its hidden layer is exponentially wide in the input dimension. Although it has been established before that such approximation can be achieved in $L_2$ sense, we prove it for $L_\infty$ approximation error and Gaussian inner weights. To the best of our knowledge, our result is the first of this kind. We give a nonasymptotic lower bound for the number of hidden layer nodes, depending on, among other things, the Lipschitz constant of the target function, the desired accuracy, and the input dimension. Our method of proof is rooted in probability theory and harmonic analysis.