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We characterize the singular values of the linear transformation associated with a standard 2D multi-channel convolutional layer, enabling their efficient computation. This characterization also leads to an algorithm for projecting a…
We consider different pentagon identities realized by the hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields…
Convergence rates of kernel density estimators for stationary time series are well studied. For invertible linear processes, we construct a new density estimator that converges, in the supremum norm, at the better, parametric, rate…
The paper deals with the construction of images from visibilities acquired using aperture synthesis instruments: Fourier synthesis, deconvolution, and spectral interpolation/extrapolation. Its intended application is to specific situations…
We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend…
The principle of translation equivariance (if an input image is translated an output image should be translated by the same amount), led to the development of convolutional neural networks that revolutionized machine vision. Other…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
A weighted space of entire functions rapidly decreasing on the real line is considered in the paper. A growth of these functions along the imaginary axis is controlled by some system of weight functions. The Fourier transform of functions…
Global convolutions have shown increasing promise as powerful general-purpose sequence models. However, training long convolutions is challenging, and kernel parameterizations must be able to learn long-range dependencies without…
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}, n=2,3,\cdots$ there…
We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…
We describe a layer-by-layer algorithm for training deep convolutional networks, where each step involves gradient updates for a two layer network followed by a simple clustering algorithm. Our algorithm stems from a deep generative model…
In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of St\"ackel equivalent systems for both degenerate and…
In this paper a new class of radial basis functions based on hyperbolic trigonometric functions will be introduced and studied. We focus on the properties of their generalised Fourier transforms with asymptotics. Therefore we will compute…
In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. Many powerful image filtering operations are based on their combinations. We establish homomorphism between max-plus semi-ring of integers…
The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this…
The article presents an algebra to represent two dimensional patterns using reciprocals of polynomials. Such a representation will be useful in neural network training and it provides a method of training patterns that is much more…
This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with…
Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. In this article we develop a novel framework for these square…
We construct a general quantization procedure for square integrable functions on well-behaved connected exponential Lie groups. The Lie groups in question should admit at least one co-adjoint orbit of maximal possible dimension. The…