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The aim of this paper is to construct a new non-uniform corner-cutting (NUCC) subdivision scheme that improves the accuracy of the classical (stationary and nonstationary) methods. The refinement rules are formulated via the reproducing…
Diagrammatic techniques to compute perturbatively the spectral properties of Euclidean Random Matrices in the high-density regime are introduced and discussed in detail. Such techniques are developed in two alternative and very different…
A new O(N) algorithm based on a recursion method, in which the computational effort is proportional to the number of atoms N, is presented for calculating the inverse of an overlap matrix which is needed in electronic structure calculations…
This paper presents an unsupervised machine learning algorithm that identifies recurring patterns -- referred to as ``music-words'' -- from symbolic music data. These patterns are fundamental to musical structure and reflect the cognitive…
In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to the square…
Unconstrained optimization problems become more common in scientific computing and engineering applications with the rapid development of artificial intelligence, and numerical methods for solving them more quickly and efficiently have been…
The high computational complexity of the multiple signal classification (MUSIC) algorithm is mainly caused by the subspace decomposition and spectrum search, especially for frequent real-time applications or massive sensors. In this paper,…
This paper proposes a numerical method using neural networks to solve the path integral problem in quantum mechanics for arbitrary potentials. The method is based on a radial basis function expansion of the interaction term that appears in…
The skeleton is an essential shape characteristic providing a compact representation of the studied shape. Its computation on the image grid raises many issues. Due to the effects of discretization, the required properties of the skeleton -…
This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective…
Recently, several instances of non-Euclidean SGD, including SignSGD, Lion, and Muon, have attracted significant interest from the optimization community due to their practical success in training deep neural networks. Consequently, a number…
Nonlinear least-squares problems are a special class of unconstrained optimization problems in which their gradient and Hessian have special structures. In this paper, we exploit these structures and proposed a matrix-free algorithm with a…
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This…
We proposed a new criterion \textit{noise-stability}, which revised the classical rigidity theory, for evaluation of MDS algorithms which can truthfully represent the fidelity of global structure reconstruction; then we proved the…
We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
The Euclidean algorithm is the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it computes a…
We give a new simple and short ("one-line") analysis for the runtime of the well-known Euclidean Algorithm. While very short simple, the obtained upper bound in near-optimal.
Multi-dimensional distributions of discrete data that resemble ellipsoids arise in numerous areas of science, statistics, and computational geometry. We describe a complete algebraic algorithm to determine the quadratic form specifying the…
In this paper, we study structured quasi-Newton methods for optimization problems with orthogonality constraints. Note that the Riemannian Hessian of the objective function requires both the Euclidean Hessian and the Euclidean gradient. In…