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This paper reviews gradient-based techniques to solve bilevel optimization problems. Bilevel optimization is a general way to frame the learning of systems that are implicitly defined through a quantity that they minimize. This…
Nonconvex optimization problems are widespread in modern machine learning and data science. We introduce an extrapolation strategy into a class of preconditioned second-order convex splitting algorithms for nonconvex optimization problems.…
We present a unified interpolation scheme that combines compactly-supported positive-definite kernels and multivariate polynomials. This unified framework generalizes interpolation with compactly-supported kernels and also classical…
We present a new open source C library \texttt{msolve} dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of \texttt{msolve} relies on Gr\''obner bases and…
For univariate polynomials over arbitrary field the degree gives an upper bound on the number of roots (factor theorem) and as a related result for any finite point-set one can construct a polynomial of degree equal to the cardinality…
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^\alpha$ for $\alpha \in…
Butterfly algorithms are an effective multilevel technique to compress discretizations of integral operators with highly oscillatory kernel functions. The particular version of the butterfly algorithm considered here realizes the transfer…
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for…
Normalization of polynomials plays a vital role in the approximate basis computation of vanishing ideals. Coefficient normalization, which normalizes a polynomial with its coefficient norm, is the most common method in computer algebra.…
The primary objective of this study is to develop novel interpolation operators that interpolate the boundary values of a function defined on a triangle. This is accomplished by constructing New Generalized Boolean sum neural network…
The performance of numerical micromagnetic models is limited by the demagnetizing field computation, which typically accounts for the majority of the computation time. For magnetization dynamics simulations explicit evaluation methods are…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum…
Polynomial multiplication is a fundamental problem in symbolic computation. There are efficient methods for the multiplication of two univariate polynomials. However, there is rarely efficiently nontrivial method for the multiplication of…
Highly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., \cite{P95}, \cite{P02}, \cite{MNP13}, and the bibliography therein), but this is still an…
An algorithm for factoring polynomials over finite fields is given by Berlekamp in 1967. The main tool was the matrix Q corresponding to each polynomial. This paper studies the degrees of polynomials over binary field that associated with…
The efficiency of Gr\"obner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
In this note we show that the degree of the interpolation polynomial for equidistant base points is characterized by the regularity of matrices of combinatorical type.
In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial…