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A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
Variational inference has recently emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) in large-scale Bayesian inference. The core idea is to trade statistical accuracy for computational efficiency. In this…
Specialized classifiers, namely those dedicated to a subset of classes, are often adopted in real-world recognition systems. However, integrating such classifiers is nontrivial. Existing methods, e.g. weighted average, usually implicitly…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
Mixed quantum-classical methods provide powerful algorithms for the simulation of quantum processes in large and complex systems. The forward-backward trajectory solution of the mixed quantum-classical Liouville equation in the mapping…
This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre…
Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v-…
In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive…
In this paper a set of previous general results for the development of B--series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B--series…
Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most…
This note provides a detailed algorithm to the application of local (perturbation) analysis of differential equations which is normally taught at graduate math courses. Exercise books often present more abstract and simplified versions of…
Modern deep learning is primarily an experimental science, in which empirical advances occasionally come at the expense of probabilistic rigor. Here we focus on one such example; namely the use of the categorical cross-entropy loss to model…
We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole…
Time series analysis is a field of data science which is interested in analyzing sequences of numerical values ordered in time. Time series are particularly interesting because they allow us to visualize and understand the evolution of a…
The remarkable generalization performance of large-scale models has been challenging the conventional wisdom of the statistical learning theory. Although recent theoretical studies have shed light on this behavior in linear models and…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
The Forward Search is an iterative algorithm for avoiding outliers in a regression analysis suggested by Hadi and Simonoff (J. Amer. Statist. Assoc. 88 (1993) 1264-1272), see also Atkinson and Riani (Robust Diagnostic Regression Analysis…
The problem of inverting a system in presence of a series-defined output is analyzed. Inverse models are derived that consist of a set of algebraic equations. The inversion is performed explicitly for an output trajectory functional, which…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…