Related papers: On the minimal algebraic complexity of the rank-on…
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…
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…
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…
In the first part of this paper we study a best approximation of a vector in Euclidean space R^n with respect to a closed semi-algebraic set C and a given semi-algebraic norm. Assuming that the given norm and its dual norm are…
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…
The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest…
We introduce a theory of relative tangency for projective algebraic varieties. The dual variety $X_Z^\vee$ of a variety $X$ relative to a subvariety $Z$ is the set of hyperplanes tangent to $X$ at a point of $Z$. We also introduce the…
We work in the space of $m$-by-$n$ real matrices with the Frobenius inner product. Consider the following Problem: Given an m-by-n real matrix A and a positive integer k, find the m-by-n matrix with rank k that is closest to A. I discuss 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 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…
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…
Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $M$, the goal is to compute a matrix $M'$ of given rank $r$ in a linear or affine…
We analyze tensors in the tensor product of three m-dimensional vector spaces satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the…
Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant…
Convex optimization over the spectrahedron, i.e., the set of all real $n\times n$ positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing and statistics, mainly as a convex…
Segre-Veronese manifolds are smooth submanifolds of tensors comprising the partially symmetric rank-1 tensors. We investigate a one-parameter family of warped geometries of Segre-Veronese manifolds, which includes the standard Euclidean…
We consider relative error low rank approximation of $tensors$ with respect to the Frobenius norm: given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+\epsilon)$OPT,…
A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products,…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
In this paper, we study linear convolutional networks with one-dimensional filters and arbitrary strides. The neuromanifold of such a network is a semialgebraic set, represented by a space of polynomials admitting specific factorizations.…