相关论文: One method for proving inequalities by computer

In this report, we aim to exemplify concentration inequalities and provide easy to understand proofs for it. Our focus is on the inequalities which are helpful in the design and analysis of machine learning algorithms.

Schemes for exact multiplication of small matrices have a large symmetry group. This group defines an equivalence relation on the set of multiplication schemes. There are algorithms to decide whether two schemes are equivalent. However, for…

This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the…

The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they have a computational advantage over some alternatives and that this makes them successful in real-world applications.…

In this paper, we introduce methods of encoding propositional logic programs in vector spaces. Interpretations are represented by vectors and programs are represented by matrices. The least model of a definite program is computed by…

We study the best approximation problem: \[ \displaystyle \min_{\alpha\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf x}_i) \right|. \] Here: $\Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\}$ is a list of…

System identification is an important area of science, which aims to describe the characteristics of the system, representing them by mathematical models. Since many of these models can be seen as recursive functions, it is extremely…

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea…

In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…

Machine learning methods often assume that the test data have the same distribution as the training data. However, this assumption may not hold due to multiple levels of heterogeneity in applications, raising issues in algorithmic fairness…

Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Such a method is introduced here, the {\em…

This paper investigates minimax quadratic programming problems with coupled inequality constraints. By leveraging a duality theorem, we develop a dual algorithm that extends the dual active set method to the minimax setting, transforming…

Orthogonalization is one of few mathematical methods conforming to mathematical standards for approximation. Finding a consistent PC matrix of a given an inconsistent PC matrix is the main goal of a pairwise comparisons method. We introduce…

In this paper, we present our ongoing work and initial results on the formal specification and verification of MiniMaple (a substantial subset of Maple with slight extensions) programs. The main goal of our work is to find behavioral errors…

This preprint is a text for students and teachers on inequalities. Some standard topics are covered on application of calculus to inequality proving. Many examples are considered, stated, solved or partially solved. Some problems are…

We present a few techniques for proving $L^p$ estimates for martingales. Basic applications to It\^o integration and rough paths are included.

In this paper, we present a rigorous framework for rational minimax approximation of matrix-valued functions that generalizes classical scalar approximation theory. Given sampled data $\{(x_\ell, {F}(x_\ell))\}_{\ell=1}^m$ where…

Equivalence between algebraic equations of motion may be detected by using a $p$-adic method, methods using factorization and linear algebra, or by systematic computer search of suitable Tschirnhausen transformations. Here, we show standard…

A new technique for approximating the entire solution set for a nonlinear system of relations (nonlinear equations, inequalities, etc. involving algebraic, smooth, or even continuous functions) is presented. The technique is to first plot…

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…