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In this article supervised learning problems are solved using soft rule ensembles. We first review the importance sampling learning ensembles (ISLE) approach that is useful for generating hard rules. The soft rules are then obtained with…

We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution…

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different…

Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously…

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of…

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…

We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact…

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…

Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods…

Foundation models often generate unreliable answers, while heuristic uncertainty estimators fail to fully distinguish correct from incorrect outputs, causing users to accept erroneous answers without any statistical guarantee. We address…

We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on…

We study the problem of evaluating the excess risk of large-scale empirical risk minimization under the square loss. Leveraging the idea of wild refitting and resampling, we assume only black-box access to the training algorithm and develop…

This work aims at solving the problems with intractable sparsity-inducing norms that are often encountered in various machine learning tasks, such as multi-task learning, subspace clustering, feature selection, robust principal component…

Estimation of actual errors from the residue in iterative solutions is necessary for efficient solution of large problems when their condition number is much larger than one. Such estimators for conjugate gradient algorithms used to solve…

In this paper, we consider the problem of column subset selection. We present a novel analysis of the spectral norm reconstruction for a simple randomized algorithm and establish a new bound that depends explicitly on the sampling…

In this paper, we consider the solution of ill-conditioned systems of linear algebraic equations that can be determined imprecisely. To improve the stability of the solution process, we "immerse" the original imprecise linear system in an…

Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend…

Estimation of the degree of stability and the bounds of solutions to non-autonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are…

The classical approach to linear system identification is given by parametric Prediction Error Methods (PEM). In this context, model complexity is often unknown so that a model order selection step is needed to suitably trade-off bias and…

We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple…