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We analyze the Brown measure the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of finitely many atoms. We use the Quaternionic Green's function, an analogue of the…
Advances in numerical optimization have supported breakthroughs in several areas of signal processing. This paper focuses on the recent enhanced variants of the proximal gradient numerical optimization algorithm, which combine quasi-Newton…
In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the…
We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various spaces of…
Gradient algorithms are classical in adaptive control and parameter estimation. For instantaneous quadratic cost functions they lead to a linear time-varying dynamic system that converges exponentially under persistence of excitation…
Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…
Mathematical models are sometime given as functions of independent input variables and equations or inequations connecting the input variables. A probabilistic characterization of such models results in treating them as functions with…
Learning algorithms related to artificial neural networks and in particular for Deep Learning may seem to involve many bells and whistles, called hyper-parameters. This chapter is meant as a practical guide with recommendations for some of…
Several quantities important in condensed matter physics, quantum information, and quantum chemistry, as well as quantities required in meta-optimization of machine learning algorithms, can be expressed as gradients of implicitly defined…
In functional data analysis (FDA), covariance function is fundamental not only as a critical quantity for understanding elementary aspects of functional data but also as an indispensable ingredient for many advanced FDA methods. This paper…
The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be…
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential…
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
Neural networks are trained primarily based on their inputs and outputs, without regard for their internal mechanisms. These neglected mechanisms determine properties that are critical for safety, like (i) transparency; (ii) the absence of…
This paper extends the notion of a p-hyponormal operator for a bounded right linear quaternionic operator defined on a right quaternionic Hilbert space. Several fundamental properties of complex p-hyponormal operators are investigated for…
Functions of several quaternion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the $\tilde \partial $-equations are studied. Moreover, quaternion Stein manifolds are…
Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear…
A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen…
Recently, the conception of slice regular functions was allowed to introduce a new quaternionic functional calculus, among which the theory of semigroups of linear operators was developed into the quaternionic setting, even in a more…
This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case.…