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Source localization and spectral estimation are among the most fundamental problems in statistical and array signal processing. Methods which rely on the orthogonality of the signal and noise subspaces, such as Pisarenko's method, MUSIC,…
Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gr\"obner bases in the 60s, there has been a lot of progress in this domain. Moreover, these…
An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is…
Depth map super-resolution (DSR) has been a fundamental task for 3D computer vision. While arbitrary scale DSR is a more realistic setting in this scenario, previous approaches predominantly suffer from the issue of inefficient…
Phase retrieval algorithms have become an important component in many modern computational imaging systems. For instance, in the context of ptychography and speckle correlation imaging, they enable imaging past the diffraction limit and…
We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such that each edge is adjacent to a previously chosen edge.…
Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step…
The real radical ideal of a system of polynomials with finitely many complex roots is generated by a system of real polynomials having only real roots and free of multiplicities. It is a central object in computational real algebraic…
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…
Loss landscape is a useful tool to characterize and compare neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly non-convex in very high dimensional space.…
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gr\"obner basis method to determine the basis for…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem.…
Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the…
An efficient method for finding all real roots of a univariate function in a given bounded domain is formulated. The proposed method uses adaptive mesh refinement to locate bracketing intervals based on bisection criterion for root finding.…
The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal…
Joint estimation of surface normals and depth is essential for holistic 3D scene understanding, yet high-resolution prediction remains difficult due to the trade-off between preserving fine local detail and maintaining global consistency.…
This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric…
Accurate interpolation of functions and derivatives is crucial in solving partial differential equations (PDEs). The Radial Basis Function (RBF) method has become an extremely popular and robust approach for interpolation on scattered data.…