arXiv subjects

Madeleine Weinstein

Publications and source records attributed to Madeleine Weinstein.

17 recordsLinked to original sources

arXiv2026

Algebraic Distance Optimization in Polyhedral Norms

We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the point and the inner normal fan of the polyhedral ball. For codimension-one varieties, we decompose $X$ into sets of points whose Voronoi cones have the same dimension, which is the expected dimension of their Voronoi cell. We prove that this decomposition is a stratification of $X$ and that each strata is a semialgebraic set. We conclude by giving an algebraic description of the medial axis, which is the locus of points whose minimal distance to $X$ is achieved at more than one point on $X$.

math.AG↗

arXiv2023

$O(k)$-Equivariant Dimensionality Reduction on Stiefel Manifolds

Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, $V_k(\mathbb{R}^N)$ and $Gr(k, \mathbb{R}^N)$ respectively, and benefit from projection onto lower-dimensional Stiefel and Grassmannian manifolds. In this work, we propose an algorithm called \textit{Principal Stiefel Coordinates (PSC)} to reduce data dimensionality from $ V_k(\mathbb{R}^N)$ to $V_k(\mathbb{R}^n)$ in an \textit{$O(k)$-equivariant} manner ($k \leq n \ll N$). We begin by observing that each element $\alpha \in V_n(\mathbb{R}^N)$ defines an isometric embedding of $V_k(\mathbb{R}^n)$ into $V_k(\mathbb{R}^N)$. Next, we describe two ways of finding a suitable embedding map $\alpha$: one via an extension of principal component analysis ($\alpha_{PCA}$), and one that further minimizes data fit error using gradient descent ($\alpha_{GD}$). Then, we define a continuous and $O(k)$-equivariant map $\pi_\alpha$ that acts as a "closest point operator" to project the data onto the image of $V_k(\mathbb{R}^n)$ in $V_k(\mathbb{R}^N)$ under the embedding determined by $\alpha$, while minimizing distortion. Because this dimensionality reduction is $O(k)$-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that $\pi_{\alpha_{PCA}}$ globally minimizes projection error in a noiseless setting, while $\pi_{\alpha_{GD}}$ achieves a meaningfully different and improved outcome when the data does not lie exactly on the image of a linearly embedded lower-dimensional Stiefel manifold as above. Multiple numerical experiments using synthetic and real-world data are performed.

cs.CG↗

arXiv2023

Voronoi Cells in Metric Algebraic Geometry of Plane Curves

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.

math.MG↗

arXiv2022

Critical Curvature of Algebraic Surfaces in Three-Space

We study the curvature of a smooth algebraic surface $X\subset \mathbb R^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number of complex critical curvature points is of order $d^3$. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.

math.AG↗

arXiv2021

Real Symmetric Matrices with Partitioned Eigenvalues

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions. For small matrices, we provide equations; for larger matrices, we explain how to use representation theory to find equations. We describe the ring of invariants under the action of the orthogonal group. For the subvariety of diagonal matrices, we give the degree.

math.AG↗

arXiv2019

The bottleneck degree of algebraic varieties

A bottleneck of a smooth algebraic variety $X \subset \mathbb{C}^n$ is a pair of distinct points $(x,y) \in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using for example numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.

math.AG↗

arXiv2019

96120: The degree of the linear orbit of a cubic surface

The projective linear group $\pgl(\comp,4)$ acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry.

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↗

arXiv2018

Voronoi Cells of Varieties

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.

math.AG↗

arXiv2018

Learning Algebraic Varieties from Samples

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.

math.AG↗

arXiv2015

Geometric-progression-free sets over quadratic number fields

A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

math.NT↗

arXiv2015

Invariance of the Sprague-Grundy Function for Variants of Wythoff's Game

We prove three conjectures of Fraenkel and Ho regarding two classes of variants of Wythoff's game. The two classes of variants of Wythoff's game feature restrictions of the diagonal moves. Each conjecture states that the Sprague-Grundy function is invariant up to a certain nim-value for a subset of that class of variant of Wythoff's game. For one class of variants of Wythoff's game, we prove that the invariance of the Sprague-Grundy function extends beyond what was conjectured by Fraenkel and Ho.

math.CO↗

arXiv2015

Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). Progression-free sets are often constructed "greedily," including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.

math.NT↗

arXiv2015

Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands and hold for more general positive linear recurrence sequences $\{G_n\}$. We generalize these results to subintervals of $[G_n, G_{n+1})$ as $n \to \infty$ for certain sequences. The analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $α(n) \to \infty$. As $n \to \infty$, for almost all $m \in [G_n, G_{n+1})$ the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals $[m, m + G_{α(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, G_{α(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval.

math.NT↗

arXiv2014

Benford Behavior of Generalized Zeckendorf Decompositions

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of non-adjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form $G_{n+1}=c_1G_n+\cdots+c_LG_{n+1-L}$ with $c_i$ positive and some other restrictions. Additionally, a set $S \subset \mathbb{Z}$ is said to satisfy Benford's law base 10 if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that as $n\to\infty$ for a randomly selected integer $m$ in $[0, G_{n+1})$ the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford's law almost surely. Our results hold more generally: one obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.

math.NT↗

arXiv2014

Benford Behavior of Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.

math.NT↗

arXiv2014

Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands. We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $α(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{α(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{α(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.

math.NT↗