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Convolutional neural networks (CNNs) have a large number of variables and hence suffer from a complexity problem for their implementation. Different methods and techniques have developed to alleviate the problem of CNN's complexity, such as…
Quantum field theory (QFT) describes nature using continuous fields, but physical properties of QFT are usually revealed in terms of measurements of observables at a finite resolution. We describe a multiscale representation of a free…
Previous research has shown that computation of convolution in the frequency domain provides a significant speedup versus traditional convolution network implementations. However, this performance increase comes at the expense of repeatedly…
Filtering of digital signals is accomplished on an Excel spreadsheet using fast Fourier transform (FFT) convolution in which the kernel is either a Gaussian or a cosine modulated Gaussian. Pedagogical examples of low-pass and band-pass…
It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique plays an important role when designing equivariant convolutions. However, the current…
Convolution operation is indispensable in studying analog optical and digital signal processing. Equally important is the correlation operation. The time domain community often teaches convolution and correlation only with one dimensional…
In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the $\theta$-translation operator, the translation operator associated with the FrFT. Subsequently,…
It has been repeatedly observed that convolutional architectures when applied to image understanding tasks learn oriented bandpass filters. A standard explanation of this result is that these filters reflect the structure of the images that…
This paper introduces versatile filters to construct efficient convolutional neural networks that are widely used in various visual recognition tasks. Considering the demands of efficient deep learning techniques running on cost-effective…
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…
The fast Fourier transform, FFT, is a useful and prevalent algorithm in signal processing. It characterizes the spectral components of a signal, or is used in combination with other operations to perform more complex computations such as…
With the use of deep learning in many areas, how to improve this technology or how to develop the structure used more effectively and in a shorter time is an issue that is of interest to many people working in this field. Many studies are…
This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network…
The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in…
This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this…
Circular convolutions and the corresponding frequency domain formula are fundamentally important in image restoration; however, in this paper, we'll prove that the usual computing method of circular convolutions violates the physical…
We present a versatile formulation of the convolution operation that we term a "mapped convolution." The standard convolution operation implicitly samples the pixel grid and computes a weighted sum. Our mapped convolution decouples these…
Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within…
In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then…
Series expansions have been a cornerstone of applied mathematics and engineering for centuries. In this paper, we revisit the Taylor series expansion from a modern Machine Learning perspective. Specifically, we introduce the Fast Continuous…