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The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter {\lambda} >= 0. Each instance (mSFN) optimizes the total weight of a homologous d-cycle and a bounded (d +…
Recently, the interpretability of deep learning has attracted a lot of attention. A plethora of methods have attempted to explain neural networks by feature visualization, saliency maps, model distillation, and so on. However, it is hard…
This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived…
In this work, we consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to…
This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…
The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes that…
This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for…
We study the size of a neural network needed to approximate the maximum function over $d$ inputs, in the most basic setting of approximating with respect to the $L_2$ norm, for continuous distributions, for a network that uses ReLU…
This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionality. The approximation results presented in this paper are…
In this paper, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our…
We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…
In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and…
Recently, the authors of \cite{SYZ22} developed a neural network with width $36d(2d + 1)$ and depth $11$, which utilizes a special activation function called the elementary universal activation function, to achieve the super approximation…
We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full…
We investigate the complexity of training a two-layer ReLU neural network with weight decay regularization. Previous research has shown that the optimal solution of this problem can be found by solving a standard cone-constrained convex…
This paper establishes an approximation theorem for randomized neural networks (RaNNs) whose hidden-layer parameters are uniformly sampled from a prescribed bounded domain. Our analysis shows that, for RaNNs of the form $\mathop{\sum}_i W_i…
Deep neural networks (DNNs) and Kolmogorov-Arnold networks (KANs) are popular methods for function approximation due to their flexibility and expressivity. However, they typically require a large number of trainable parameters to produce a…
This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor…
This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the…