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Miruna-Stefana Sorea

Publications and source records attributed to Miruna-Stefana Sorea.

18 recordsLinked to original sources
arXiv2025

Tropical Bisectors and Carlini-Wagner Attacks

Pasque et al. showed that using a tropical symmetric metric as an activation function in the last layer can improve the robustness of convolutional neural networks (CNNs) against state-of-the-art attacks, including the Carlini-Wagner attack. This improvement occurs when the attacks are not specifically adapted to the non-differentiability of the tropical layer. Moreover, they showed that the decision boundary of a tropical CNN is defined by tropical bisectors. In this paper, we explore the combinatorics of tropical bisectors and analyze how the tropical embedding layer enhances robustness against Carlini-Wagner attacks. We prove an upper bound on the number of linear segments the decision boundary of a tropical CNN can have. We then propose a refined version of the Carlini-Wagner attack, specifically tailored for the tropical architecture. Computational experiments with MNIST and LeNet5 showcase our attacks improved success rate.

cs.LG↗
arXiv2024

Adjoints and Canonical Forms of Polypols

Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.

math.AG↗
arXiv2023

Exact Solutions in Log-Concave Maximum Likelihood Estimation

We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized W-Lambert function. Even more, we show that finding the log-concave maximum likelihood estimate is equivalent to solving a collection of polynomial-exponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's alpha-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation.

math.ST↗
arXiv2023

The toric locus of a reaction network is a smooth manifold

We show that the toric locus of a reaction network is a smoothly embedded submanifold of the Euclidean space. More precisely, we prove that the toric locus of a reaction network is the image of an embedding and it is diffeomorphic to the product space between the affine invariant polyhedron of the network and its set of complex-balanced flux vectors. Moreover, we prove that within each affine invariant polyhedron, the complex-balanced equilibrium depends smoothly on the parameters (i.e., reaction rate constants). We also show that the complex-balanced equilibrium depends smoothly on the initial conditions.

math.DS↗
arXiv2023

The disguised toric locus and affine equivalence of reaction networks

Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.

math.DS↗
arXiv2023

Poincaré-Reeb graphs of real algebraic domains

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincaré-Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré-Reeb graph.

math.GT↗
arXiv2023

The structure of the moduli space of toric dynamical systems of a reaction network

We consider toric dynamical systems, which are also called complex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under the assumption of mass-action kinetics, they can give rise to complex-balanced equilibria. Given a reaction network, we study the moduli space of toric dynamical systems generated by this network, also called the toric locus of the network. The toric locus is an algebraic variety, and we are especially interested in its topological properties. We show that complex-balanced equilibria depend continuously on the parameter values in the toric locus, and, using this result, we prove that the toric locus has a remarkable product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron of the network. In particular, it follows that the toric locus is a contractible manifold. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.

math.AG↗
arXiv2023

The structure of the moduli spaces of toric dynamical systems

We consider complex-balanced mass-action systems, or toric dynamical systems. They are remarkably stable polynomial dynamical systems arising from reaction networks seen as Euclidean embedded graphs. We study the moduli spaces of toric dynamical systems, called the toric locus: given a reaction network, we are interested in the topological structure of the set of parameters giving rise to toric dynamical systems. First we show that the complex-balanced equilibria depend continuously on the parameter values. Using this result, we prove that the toric locus of any toric dynamical system is connected. In particular, we emphasize its product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.

math.DS↗
arXiv2022

Symmetrically colored Gaussian graphical models with toric vanishing ideal

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras.

math.CO↗
arXiv2022

Families of polytopes with rational linear precision in higher dimensions

In this article we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile's theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.

math.CO↗
arXiv2022

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.

math.AG↗
arXiv2021

A generalization of the space of complete quadrics

To any homogeneous polynomial $h$ we naturally associate a variety $Ω_h$ which maps birationally onto the graph $Γ_h$ of the gradient map $\nabla h$ and which agrees with the space of complete quadrics when $h$ is the determinant of the generic symmetric matrix. We give a sufficient criterion for $Ω_h$ being smooth which applies for example when $h$ is an elementary symmetric polynomial. In this case $Ω_h$ is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when $Ω_h$ is not smooth.

math.AG↗
arXiv2020

On the eigenpoints of cubic surfaces

We show that the eigenschemes of $4 \times 4 \times 4$ symmetric tensors are parametrized by a linear subvariety of the Grassmannian $\operatorname{Gr}(3,\mathbb{P}^{14})$. We also study the decomposition of the eigenscheme into the subscheme associated to the zero eigenvalue and its residue. In particular, we categorize the possible degrees and dimensions.

math.AG↗
arXiv2019

Permutations encoding the local shape of level curves of real polynomials via generic projections

The non-convexity of a smooth and compact connected component of a real algebraic plane curve can be measured by a combinatorial object called the Poincare-Reeb tree associated to the curve and to a direction of projection. In this paper we show that if the chosen projection avoids the bitangents and the inflectional tangencies to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincare-Reeb tree becomes a complete binary tree and its vertices become endowed with a total order relation. Such a projection direction is called generic. We prove that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the choice of the direction of projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, that we call snakes. The snakes offer an effective description of the local geometry and topology, well-suited for further computations.

math.AG↗
arXiv2019

Measuring the local non-convexity of real algebraic curves

The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin of the real affine plane. We introduce and describe a new combinatorial object, called the Poincare-Reeb graph, whose role is to encode the shape of such curves and to allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincare-Reeb graph is a plane tree and can be used as a tool to study the asymptotic behaviour of level curves near a strict local minimum. Finally, using the real polar curve, we show that locally the shape of the levels stabilises and that no spiralling phenomena occur near the origin.

math.AG↗
arXiv2019

Constructing Separable Arnold Snakes of Morse Polynomials

We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values are distinct. Thus we can associate to it an alternating permutation: the so-called Arnold snake, given by the relative positions of its critical values. We realise any separable alternating permutation as the Arnold snake of a Morse polynomial.

math.AG↗