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Amy Wiebe

Publications and source records attributed to Amy Wiebe.

17 recordsLinked to original sources
arXiv2026

From Design Principles to Prototype: A Game for Students with ADHD and Learning Disabilities Transitioning to Post-Secondary Education

Students with Attention Deficit Hyperactivity Disorder (ADHD) and Learning Disabilities (LD) can face significant academic, social, and organizational challenges when transitioning to post-secondary education. This paper presents a literature-informed serious game prototype designed to support this transition. We synthesize prior work into design considerations for students with ADHD and LD and show how these considerations are instantiated in a story-driven game.

cs.MM↗
arXiv2026

Matrices of nonnegative integer rank two

The nonnegative integer rank of a matrix is a variant of the classical nonnegative rank, introduced in the 1980s, where factorizations are required to have integer entries. While computing nonnegative integer rank is generally very hard, we focus on a fundamental special case: determining when a rank 2 nonnegative integer matrix has nonnegative integer rank equal to 2 (the "rank2 problem"). Although this problem is trivial in the continuous case, in this context it is surprisingly rich. We provide a geometric reformulation in terms of affine semigroups and rational cones in the plane, which yields new structural insights. We show that any rank 2 integer matrix can be reduced to a $3\times 3$ matrix which has nonnegative integer rank $2$ if and only if the original one also has nonnegative integer rank $2$, with the reduction computable in polynomial time. This reduction reveals that the difficulty of the rank2 problem is already captured by small matrices. Building on this geometric framework, we also develop an algorithm that solves the rank2 problem by strategically searching for integer generators within bounded regions of the associated cone. Although the theoretical worst-case complexity remains high, numerical experiments demonstrate that the algorithm performs efficiently in practice.

math.CO↗
arXiv2025

CLARSTA: A random subspace trust-region algorithm for convex-constrained derivative-free optimization

This paper proposes a random subspace trust-region algorithm for general convex-constrained derivative-free optimization (DFO) problems. Similar to previous random subspace DFO methods, the convergence of our algorithm requires a certain accuracy of models and a certain quality of subspaces. For model accuracy, we define a new class of models that is only required to provide reasonable accuracy on the projection of the constraint set onto the subspace. We provide a new geometry measure to make these models easy to analyze, construct, and manage. For subspace quality, we use a normalized random orthogonal projection matrix to provide a method to sample subspaces that preserve the first-order criticality measure by a certain fraction with a probability lower bound independent of the ambient dimension. Based on all these new theoretical results, we present an almost-sure liminf convergence result and an $\mathcal{O}(\epsilon^{-2})$ expected iteration complexity analysis of our algorithm. Numerical experiments on problems with dimensions up to 10000 demonstrate the reliable performance of our algorithm in high dimensions.

math.OC↗
arXiv2025

Determining inscribability of polytopes via rank minimization based on slack matrices

A polytope is inscribable if there is a realization where all vertices lie on the sphere. In this paper, we provide a necessary and sufficient condition for a polytope to be inscribable. Based on this condition, we characterize the problem of determining inscribability as a minimum rank optimization problem using slack matrices. We propose an SDP approximation for the minimum rank optimization problem and prove that it is tight for certain classes of polytopes. Given a polytope, we provide three algorithms to determine its inscribability. All the optimization problems and algorithms we propose in this paper depend on the number of vertices and facets but are independent of the dimension of the polytope. Numerical results demonstrate our SDP approximation's efficiency, accuracy, and robustness for determining inscribability of simplicial polytopes of dimensions $4\le d\le 8$ with vertices $n\le 10$, revealing its potential in high dimensions.

math.CO↗
arXiv2024

A Yannakakis-type theorem for lifts of affine semigroups

Yannakakis' theorem relating the extension complexity of a polytope to the size of a nonnegative factorization of its slack matrix is a seminal result in the study of lifts of convex sets. Inspired by this result and the importance of lifts in the setting of integer programming, we show that a similar result holds for the discrete analog of convex polyhedral cones-affine semigroups. We define the notions of the integer slack matrix and a lift of an affine semigroup. We show that many of the characterizations of the slack matrix in the convex cone setting have analogous results in the affine semigroup setting. We also show how slack matrices of affine semigroups can be used to obtain new results in the study of nonnegative integer rank of nonnegative integer matrices.

math.CO↗
arXiv2024

A general framework for floating point error analysis of simplex derivatives

Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient, centred simplex gradient, and adapted centred simplex gradient, are in the form of simplex derivatives. Owing to machine precision, the approximation accuracy of any numerical approximation technique is subject to the influence of floating point errors. In this paper, we provide a general framework for floating point error analysis of simplex derivatives. Our framework is independent of the choice of the simplex derivative as long as it satisfies a general form. We review the definition and approximation accuracy of the generalized simplex gradient and generalized centred simplex gradient. We define and analyze the accuracy of a generalized version of the adapted centred simplex gradient. As examples, we apply our framework to the generalized simplex gradient, generalized centred simplex gradient, and generalized adapted centred simplex gradient. Based on the results, we give suggestions on the minimal choice of approximate diameter of the sample set.

math.NA↗
arXiv2023

$Q$-fully Quadratic Modeling and its Application in a Random Subspace Derivative-free Method

Model-based derivative-free optimization (DFO) methods are an important class of DFO methods that are known to struggle with solving high-dimensional optimization problems. Recent research has shown that incorporating random subspaces into model-based DFO methods has the potential to improve their performance on high-dimensional problems. However, most of the current theoretical and practical results are based on linear approximation models due to the complexity of quadratic approximation models. This paper proposes a random subspace trust-region algorithm based on quadratic approximations. Unlike most of its precursors, this algorithm does not require any special form of objective function. We study the geometry of sample sets, the error bounds for approximations, and the quality of subspaces. In particular, we provide a technique to construct $Q$-fully quadratic models, which is easy to analyze and implement. We present an almost-sure global convergence result of our algorithm and give an upper bound on the expected number of iterations to find a sufficiently small gradient. We also develop numerical experiments to compare the performance of our algorithm using both linear and quadratic approximation models. The numerical results demonstrate the strengths and weaknesses of using quadratic approximations.

math.OC↗
arXiv2021

General non-realizability certificates for spheres with linear programming

In this paper we present a simple technique to derive certificates of non-realizability for an abstract polytopal sphere. Our approach uses a variant of the classical algebraic certificates introduced by Bokowski and Sturmfels in [Computational Synthetic Geometry, 1989], the final polynomials. More specifically we reduce the problem of finding a realization to that of finding a positive point in a variety and try to find a polynomial with positive coefficients in the generating ideal (a positive polynomial), showing that such point does not exist. Many, if not most, of the techniques for proving non-realizability developed in the last three decades can be seen as following this framework, using more or less elaborate ways of constructing such positive polynomials. Our proposal is more straightforward as we simply use linear programming to exhaustively search for such positive polynomials in the ideal restricted to some linear subspace. Somewhat surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives, and allows us to derive new examples of non-realizable abstract polytopal spheres.

math.CO↗
arXiv2020

Combining realization space models of polytopes

In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of [Criado, Santos. Topological prismatoids and small simplicial spheres of large diameter. Experimental Mathematics, 1-13, 2019], about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them.

math.CO↗
arXiv2020

Slack Ideals in Macaulay2

Recently Gouveia, Thomas and the authors introduced the slack realization space, a new model for the realization space of a polytope. It represents each polytope by its slack matrix, the matrix obtained by evaluating each facet inequality at each vertex. Unlike the classical model, the slack model naturally mods out projective transformations. It is inherently algebraic, arising as the positive part of a variety of a saturated determinantal ideal, and provides a new computational tool to study classical realizability problems for polytopes. We introduce the package SlackIdeals for Macaulay2, that provides methods for creating and manipulating slack matrices and slack ideals of convex polytopes and matroids. Slack ideals are often difficult to compute. To improve the power of the slack model, we develop two strategies to simplify computations: we scale as many entries of the slack matrix as possible to one; we then obtain a reduced slack model combining the slack variety with the more compact Grassmannian realization space model. This allows us to study slack ideals that were previously out of computational reach. As applications, we show that the well-known Perles polytope does not admit rational realizations and prove the non-realizability of a large quasi-simplicial sphere.

math.CO↗
arXiv2020

Combinatorial Methods for Minkowski Tensors of Polytopes

In this paper we use a generating function approach to record and calculate entries of the Minkowski tensors of a polytope. We focus on ''surface tensors'', extending the methods used in arXiv:1807.10258 for moments of the uniform distribution which correspond to volume tensors. In this context we also extend the definition of the adjoint polynomial to the boundary complex of a polytope with simplicial facets. In the case of simplicial polytopes we give an explicit formulation for these surface tensors.

math.CO↗
arXiv2019

The Slack Realization Space of a Polytope

In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes.

math.CO↗
arXiv2019

Projectively unique polytopes and toric slack ideals

The slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. The simplest slack ideals are toric and have connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non-incidences of the polytope. The slack ideal of a polytope is contained in this toric ideal if and only if the polytope is morally 2-level, a generalization of the 2-level property in polytopes. We show that polytopes that do not admit rational realizations cannot have toric slack ideals. A classical example of a projectively unique polytope with no rational realizations is due to Perles. We prove that the slack ideal of the Perles polytope is reducible, providing the first example of a slack ideal that is not prime.

math.CO↗
arXiv2018

The slack realization space of a matroid

We introduce a new model for the realization space of a matroid, which is obtained from a variety defined by a saturated determinantal ideal, called the slack ideal, coming from the vertex-hyperplane incidence matrix of the matroid. This is inspired by a similar model for the slack realization space of a polytope. We show how to use these ideas to certify non-realizability of matroids, and describe an explicit relationship to the standard Grassmann-Plücker realization space model. We also exhibit a way of detecting projectively unique matroids via their slack ideals by introducing a toric ideal that can be associated to any matroid.

math.CO↗
arXiv2015

Constructions of complex equiangular lines from mutually unbiased bases

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: (1) adapting a set of $d$ MUBs in $\mathbb{C}^d$ to obtain $d^2$ equiangular lines in $\mathbb{C}^d$, (2) using a set of $d$ MUBs in $\mathbb{C}^d$ to build $(2d)^2$ equiangular lines in $\mathbb{C}^{2d}$, (3) combining two copies of a set of $d$ MUBs in $\mathbb{C}^d$ to build $(2d)^2$ equiangular lines in $\mathbb{C}^{2d}$. For each construction, we give the dimensions $d$ for which we currently know that the construction produces a maximum-sized set of equiangular lines.

math.CO↗
arXiv2015

A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.

math.CO↗