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For years, many neural networks have been developed based on the Kolmogorov-Arnold Representation Theorem (KART), which was created to address Hilbert's 13th problem. Recently, relying on KART, Kolmogorov-Arnold Networks (KANs) have…

We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions -- a Linear/Ridge expansion. For an explicitly or implicitly given function, we reformulate finding a best Linear/Ridge…

We propose a method to efficiently compute tomographic projections of a 3D volume represented by a linear combination of shifted B-splines. To do so, we propose a ray-tracing algorithm that computes 3D line integrals with arbitrary…

Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods…

In Masked Image Modeling (MIM), two primary methods exist: Pixel MIM and Latent MIM, each utilizing different reconstruction targets, raw pixels and latent representations, respectively. Pixel MIM tends to capture low-level visual details…

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…

We present a new simple method for rounding a semidefinite programming relaxation of a constraint satisfaction problem. We apply it to the problem of approximate angular synchronization. Specifically, we are given directed distances on a…

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far…

Although image super-resolution (SR) problem has experienced unprecedented restoration accuracy with deep neural networks, it has yet limited versatile applications due to the substantial computational costs. Since different input images…

We obtain new complexity bounds for computing a triangular integral basis of a number field or a function field. We reach for function fields a softly linear cost with respect to the size of the output when the residual characteristic is…

A fundamental problem in machine learning is to understand how neural networks make accurate predictions, while seemingly bypassing the curse of dimensionality. A possible explanation is that common training algorithms for neural networks…

This paper deals with robust regression and subspace estimation and more precisely with the problem of minimizing a saturated loss function. In particular, we focus on computational complexity issues and show that an exact algorithm with…

We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple…

In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…

We present a fast and simple technique to convert images into a radiance surface-based scene representation. Building on existing radiance volume reconstruction algorithms, we introduce a subtle yet impactful modification of the loss…

Feynman integral reduction by means of integration-by-parts identities is a major power gadget in a theorist toolbox indispensable for calculation of multiloop quantum effects relevant for particle phenomenology and formal theory alike. An…

In robotic vision, a de-facto paradigm is to learn in simulated environments and then transfer to real-world applications, which poses an essential challenge in bridging the sim-to-real domain gap. While mainstream works tackle this problem…

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…

We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm…

In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…