This paper presents a new approach of constructing $α$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $α$-fractals utilizing neural network-based operators, providing an approach to generating fractal functions with interpolation properties. Based on the same foundation, we have developed fractal interpolation functions that utilize only the values of the original function at the nodes or partition points, unlike traditional methods that rely on the entire original function. Further, we have constructed \(α\)-fractals that preserve the smoothness of functions under certain constraints by employing a four-layered neural network operator, ensuring that if \(f \in C^{r}[a,b]\), then the corresponding fractal \(f^α \in C^{r}[a,b]\). Furthermore, we analyze the convergence of these $α$-fractals to the original function under suitable conditions. The work uses key approximation theory tools, such as the modulus of continuity and interpolation operators, to develop convergence results and uniform approximation error bounds.

This paper develops a new data-dependent output activation function base on interpolation function. It is a nonparametric model based on a subset of training data. The activation function is defined in an implicit manner by solving a set of linear equations. Therefore, it cannot be solved directly by backpropagation. Instead it proposes an auxiliary network with linear output to approximate the gradient.

Continuous-time trajectory estimation is an attractive alternative to discrete-time batch estimation due to the ability to incorporate high-frequency measurements from asynchronous sensors while keeping the number of optimization parameters bounded. Two types of continuous-time estimation have become prevalent in the literature: Gaussian process regression and spline-based estimation. In this paper, we present a direct comparison between these two methods. We first compare them using a simple linear system, and then compare them in a camera and IMU sensor fusion scenario on SE(3) in both simulation and hardware. Our results show that if the same measurements and motion model are used, the two methods achieve similar trajectory accuracy. In addition, if the spline order is chosen so that the degree-of-differentiability of the two trajectory representations match, then they achieve similar solve times as well.

Dilated Convolution with Learnable Spacings (DCLS) is a recently proposed variation of the dilated convolution in which the spacings between the non-zero elements in the kernel, or equivalently their positions, are learnable. Non-integer positions are handled via interpolation. Thanks to this trick, positions have well-defined gradients. The original DCLS used bilinear interpolation, and thus only considered the four nearest pixels. Yet here we show that longer range interpolations, and in particular a Gaussian interpolation, allow improving performance on ImageNet1k classification on two state-of-the-art convolutional architectures (ConvNeXt and Conv\-Former), without increasing the number of parameters. The method code is based on PyTorch and is available at https://github.com/K-H-Ismail/Dilated-Convolution-with-Learnable-Spacings-PyTorch

We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.

Modifiable combining functions are a synthesis of two general approaches to combining evidence. Because they facilitate the acquisition, representation, explanation, and modification of expert knowledge about combinations of evidence, they are presented as a device for knowledge engineers, not as a normative theory of evidence combination. The basic idea of modifiable combining functions is to acquire degrees of belief for a subset of all possible combinations of evidence, then infer degrees of belief for other combinations in the set.

This paper investigates spatiotemporal interpolation methods for the application of air pollution assessment. The air pollutant of interest in this paper is fine particulate matter PM2.5. The choice of the time scale is investigated when applying the shape function-based method. It is found that the measurement scale of the time dimension has an impact on the interpolation results. Based upon the comparison between the accuracies of interpolation results, the most effective time scale out of four experimental ones was selected for performing the PM2.5 interpolation. The paper also evaluates the population exposure to the ambient air pollution of PM2.5 at the county-level in the contiguous U.S. in 2009. The interpolated county-level PM2.5 has been linked to 2009 population data and the population with a risky PM2.5 exposure has been estimated. The risky PM2.5 exposure means the PM2.5 concentration exceeding the National Ambient Air Quality Standards. The geographic distribution of the counties with a risky PM2.5 exposure is visualized. This work is essential to understanding the associations between ambient air pollution exposure and population health outcomes.