Related papers: The Euclidean Distance Degree of Fermat Hypersurfa…
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value…
In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree…
We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the…
The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry. It has direct applications in geometric modeling, computer vision, and statistics. We use non-proper Morse theory to give a…
We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in $\Bbb C^n$. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a…
We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely…
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a…
Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of…
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…
We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable…
We initiate a study of the Euclidean Distance Degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general given its support, such that the support contains the origin.…
Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the…
Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We give a formula of the Maximum likelihood…
A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…
We study an optimization problem with the feasible set being a real algebraic variety $X$ and whose parametric objective function $f_u$ is gradient-solvable with respect to the parametric data $u$. This class of problems includes Euclidean…
We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in…
We study the Hermitian distance degree, a real enumerative invariant counting critical points of the squared Hermitian distance function, for matrix varieties invariant under left and right unitary actions. For such a variety \(M \subset…
Computing the Fr\'{e}chet distance for surfaces is a surprisingly hard problem and the only known algorithm is limited to computing it between flat surfaces. We adapt this algorithm to create one for computing the Fr\'{e}chet distance for a…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…