arXiv subjects

Emil Horobet

Publications and source records attributed to Emil Horobet.

13 recordsLinked to original sources

arXiv2025

The geometry of higher order modern portfolio theory

In this article, we study the generalized modern portfolio theory, with utility functions admitting higher-order cumulants. We establish that under certain genericity conditions, the utility function has a constant number of complex critical points. We study the discriminant locus of complex critical points with multiplicity. Finally, we switch our attention to the generalization of the feasible portfolio set (variety), determine its dimension, and give a formula for its degree.

q-fin.PM↗

arXiv2025

Infinite Euclidean Distance Discriminants

We study infinite Euclidean distance discriminants of algebraic varieties, defined as the loci of data points whose fibers under the second projection from the Euclidean distance correspondence are positive-dimensional. In particular, these discriminants contain all data points with infinitely many critical points for the nearest-point problem. We present computer code that computes the infinite Euclidean distance discriminant, and use it to present numerous varieties with nonempty such discriminants. Moreover, we prove that for any data point, the fiber under the second projection is contained in a finite union of hyperspheres centered at that point. For curves, we include a complete characterization; their infinite Euclidean distance discriminants turn out to be affine linear spaces. Finally, we introduce and characterize skew-tube surfaces in three-dimensional space. By construction, these have a one-dimensional infinite Euclidean distance discriminant. We further demonstrate that many skew-tubes have significantly lower Euclidean distance degrees than generic surfaces of the same degree.

math.AG↗

arXiv2024

Decomposing tensors via rank-one approximations

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors.

math.AG↗

arXiv2023

When does subtracting a rank-one approximation decrease tensor rank?

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

math.AG↗

arXiv2023

The critical curvature degree of an algebraic variety

In this article, we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing it.

math.AG↗

arXiv2022

Data loci in algebraic optimization

In this article we provide examples, methods and algorithms to determine conditions on the parameters of certain type of parametric optimization problems, such that among the resulting local minima and maxima there is at least one which satisfies given polynomial conditions (for example it is singular or symmetric).

math.AG↗

arXiv2019

Offset Hypersurfaces and Persistent Homology of Algebraic Varieties

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 algebraic optimization. Namely, we express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean Distance Degree of the starting variety, obtaining a new way to compute these degrees. Finally, we describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.

math.AG↗

arXiv2015

Orthogonal and unitary tensor decomposition from an algebraic perspective

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras---associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.

math.AG↗

arXiv2015

The average number of critical rank-one approximations to a tensor

Motivated by the many potential applications of low-rank multi-way tensor approximations, we set out to count the rank-one tensors that are critical points of the distance function to a general tensor v. As this count depends on v, we average over v drawn from a Gaussian distribution, and find formulas that relates this average to problems in random matrix theory.

math.OC↗

arXiv2015

The Maximum Likelihood Data Singular Locus

For general data, the number of complex solutions to the likelihood equations is constant and this number is called the (maximum likelihood) ML-degree of the model. In this article, we describe the special locus of data for which the likelihood equations have a solution in the model's singular locus.

math.AG↗

arXiv2015

The Data Singular and the Data Isotropic Loci for Affine Cones

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree. The two special loci of the data points where the number of critical points is smaller then the ED degree are called the Euclidean Distance Data Singular Locus and the Euclidean Distance Data Isotropic Locus. In this article we present connections between these two special loci of an affine cone and its dual cone.

math.AG↗

arXiv2014

Algebraic boundary of matrices of nonnegative rank at most three

The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

math.AG↗

arXiv2014

The Euclidean distance degree of an algebraic variety

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 decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

math.AG↗