Related papers: Beyond Linear Algebra
By virtue of its great utility in solving real-world problems, optimization modeling has been widely employed for optimal decision-making across various sectors, but it requires substantial expertise from operations research professionals.…
Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
We study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when the dataset size is small with respect to the model, and is…
In this article, we discuss some of the recent developments in applying machine learning (ML) techniques to nonlinear dynamical systems. In particular, we demonstrate how to build a suitable ML framework for addressing two specific…
We consider a general class of non-linear Bellman equations. These open up a design space of algorithms that have interesting properties, which has two potential advantages. First, we can perhaps better model natural phenomena. For…
It seems that in the current age, computers, computation, and data have an increasingly important role to play in scientific research and discovery. This is reflected in part by the rise of machine learning and artificial intelligence,…
Linear systems under the influence of nonlinear and random linear perturbations, and with random initial and boundary conditions, are discussed. The notion of states of a system is substituted by the notion of the generating vectors for…
The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of…
In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems…
In this paper, we address a class of specially structured problems that include speed planning, for mobile robots and robotic manipulators, and dynamic programming. We develop two new numerical procedures, that apply to the general case and…
Continual Learning (CL) focuses on maximizing the predictive performance of a model across a non-stationary stream of data. Unfortunately, CL models tend to forget previous knowledge, thus often underperforming when compared with an offline…
In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly…
The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the…
This monograph is about discrete energy minimization for discrete graphical models. It considers graphical models, or, more precisely, maximum a posteriori inference for graphical models, purely as a combinatorial optimization problem.…
On one hand, termination analysis of logic programs is now a fairly established research topic within the logic programming community. On the other hand, non-termination analysis seems to remain a much less attractive subject. If we divide…
Linear principal component analysis (PCA) can be extended to a nonlinear PCA by using artificial neural networks. But the benefit of curved components requires a careful control of the model complexity. Moreover, standard techniques for…
Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
The standard methodology handling nonlinear PDE's involves the two steps: numerical discretization to get a set of nonlinear algebraic equations, and then the application of the Newton iterative linearization or its variants to solve the…