arXiv subjects

Chris Peterson

Publications and source records attributed to Chris Peterson.

At least 19 recordsLinked to original sources

arXiv2026

Weak and strong Lefschetz properties for Hartshorne-Rao modules of curves in $\mathbb P^3$

Let $C\subset \mathbb P^3$ be a curve over an algebraically closed field of characteristic zero, and let $M(C)$ denote its Hartshorne-Rao module. We study how the geometry of $C$ influences whether $M(C)$ satisfies the Weak and Strong Lefschetz Properties. We first consider unions of general skew lines and prove that multiplication by $L^i$, for a general linear form $L$, has maximal rank on $M(C)$ for $i=1,2,3$. The proof uses a specialization to zero-dimensional schemes that can be written as a union of curvilinear schemes, each of a particular type and of degree at most three, together with generic Hilbert function results for such schemes, which are of independent interest. We then examine how special geometric configurations can affect the Weak Lefschetz Property. In particular, we show that curves on a smooth quadric surface have Hartshorne-Rao modules with the Weak Lefschetz Property, and that the property persists for unions of skew lines with all but one line on a quadric. By contrast, for $r\geq 10$, we construct configurations of $r$ skew lines with all but two lines on a quadric whose Hartshorne-Rao modules fail the Weak Lefschetz Property. Finally, we study smooth irreducible curves. We prove the Weak Lefschetz Property in several low-degree cases, construct a degree 15 curve for which it fails, and show that general nondegenerate rational curves have Hartshorne-Rao modules with the Weak Lefschetz Property. These results illustrate both the strength and the limitations of geometric hypotheses in controlling Lefschetz properties of Hartshorne-Rao modules.

math.AG↗

arXiv2025

A Granular Grassmannian Clustering Framework via the Schubert Variety of Best Fit

In many classification and clustering tasks, it is useful to compute a geometric representative for a dataset or a cluster, such as a mean or median. When datasets are represented by subspaces, these representatives become points on the Grassmann or flag manifold, with distances induced by their geometry, often via principal angles. We introduce a subspace clustering algorithm that replaces subspace means with a trainable prototype defined as a Schubert Variety of Best Fit (SVBF) - a subspace that comes as close as possible to intersecting each cluster member in at least one fixed direction. Integrated in the Linde-Buzo-Grey (LBG) pipeline, this SVBF-LBG scheme yields improved cluster purity on synthetic, image, spectral, and video action data, while retaining the mathematical structure required for downstream analysis.

cs.LG↗

arXiv2025

Oriented Steiner Triple Systems, Steiner Products, and Dynamics

Let S denote a Steiner triple system on an n-element set. An orientation of S is an assignment of a cyclic ordering to each of the triples in S. From an oriented Steiner triple system, one can define an anticommutative bilinear operation on Rn resembling the cross product. We call this bilinear operation a Steiner product. We classify the oriented Steiner triple systems on sets of size 7 and 9 and investigate the dynamics of their associated Steiner products.

math.CO↗

arXiv2023

ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.

math.AT↗

arXiv2022

Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity $m$ in terms of their $m$-Apéry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with $n$ sporadic elements and Frobenius number $f$ is polynomial up to periodicity, or quasi-polynomial, as a function of $f$ for fixed $n$. We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope.

math.CO↗

arXiv2022

The Flag Median and FlagIRLS

Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust representations for datasets of images, videos and more. Since subspaces correspond to points on a Grassmann manifold, one is led to consider the idea of a subspace prototype for a Grassmann-valued dataset. While a number of different subspace prototypes have been described, the calculation of some of these prototypes has proven to be computationally expensive while other prototypes are affected by outliers and produce highly imperfect clustering on noisy data. This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like Linde-Buzo-Grey (LBG) to produce improved clusterings on Grassmannians. Numerical experiments include a synthetic dataset, the MNIST handwritten digits dataset, the Mind's Eye video dataset and the UCF YouTube action dataset. The flag median is compared the other leading algorithms for computing prototypes on the Grassmannian, namely, the $\ell_2$-median and to the flag mean. We find that using FlagIRLS to compute the flag median converges in $4$ iterations on a synthetic dataset. We also see that Grassmannian LBG with a codebook size of $20$ and using the flag median produces at least a $10\%$ improvement in cluster purity over Grassmannian LBG using the flag mean or $\ell_2$-median on the Mind's Eye dataset.

stat.ML↗

arXiv2021

Supporting Massive DLRM Inference Through Software Defined Memory

Deep Learning Recommendation Models (DLRM) are widespread, account for a considerable data center footprint, and grow by more than 1.5x per year. With model size soon to be in terabytes range, leveraging Storage ClassMemory (SCM) for inference enables lower power consumption and cost. This paper evaluates the major challenges in extending the memory hierarchy to SCM for DLRM, and presents different techniques to improve performance through a Software Defined Memory. We show how underlying technologies such as Nand Flash and 3DXP differentiate, and relate to real world scenarios, enabling from 5% to 29% power savings.

cs.AR↗

arXiv2020

Locally Linear Attributes of ReLU Neural Networks

A ReLU neural network determines/is a continuous piecewise linear map from an input space to an output space. The weights in the neural network determine a decomposition of the input space into convex polytopes and on each of these polytopes the network can be described by a single affine mapping. The structure of the decomposition, together with the affine map attached to each polytope, can be analyzed to investigate the behavior of the associated neural network.

cs.LG↗

arXiv2020

Metric thickenings and group actions

Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and Čech complexes of $X/G$. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris-Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.

math.MG↗

arXiv2020

The flag manifold as a tool for analyzing and comparing data sets

The shape and orientation of data clouds reflect variability in observations that can confound pattern recognition systems. Subspace methods, utilizing Grassmann manifolds, have been a great aid in dealing with such variability. However, this usefulness begins to falter when the data cloud contains sufficiently many outliers corresponding to stray elements from another class or when the number of data points is larger than the number of features. We illustrate how nested subspace methods, utilizing flag manifolds, can help to deal with such additional confounding factors. Flag manifolds, which are parameter spaces for nested subspaces, are a natural geometric generalization of Grassmann manifolds. To make practical comparisons on a flag manifold, algorithms are proposed for determining the distances between points $[A], [B]$ on a flag manifold, where $A$ and $B$ are arbitrary orthogonal matrix representatives for $[A]$ and $[B]$, and for determining the initial direction of these minimal length geodesics. The approach is illustrated in the context of (hyper) spectral imagery showing the impact of ambient dimension, sample dimension, and flag structure.

cs.CV↗

arXiv2020

Distributions of Distances and Volumes of Balls in Homogeneous Lens Spaces

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the $k$th moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.

math.DG↗

arXiv2020

The apolar algebra of a product of linear forms

Apolarity is an important tool in commutative algebra and algebraic geometry which studies a form, $f$, by the action of polynomial differential operators on $f$. The quotient of all polynomial differential operators by those which annihilate $f$ is called the apolar algebra of $f$. In general, the apolar algebra of a form is useful for determining its Waring rank, which can be seen as the problem of decomposing the supersymmetric tensor, associated to the form, minimally as a sum of rank one supersymmetric tensors. In this article we study the apolar algebra of a product of linear forms, which generalizes the case of monomials and connects to the geometry of hyperplane arrangements. In the first part of the article we provide a bound on the Waring rank of a product of linear forms under certain genericity assumptions; for this we use the defining equations of so-called star configurations due to Geramita, Harbourne, and Migliore. In the second part of the article we use the computer algebra system Bertini, which operates by homotopy continuation methods, to solve certain rank equations for catalecticant matrices. Our computations suggest that, up to a change of variables, there are exactly six homogeneous polynomials of degree six in three variables which factor completely as a product of linear forms defining an irreducible multi-arrangement and whose apolar algebras have dimension six in degree three. As a consequence of these calculations, we find six cases of such forms with cactus rank six, five of which also have Waring rank six. Among these are products defining subarrangements of the braid and Hessian arrangements.

math.AC↗

arXiv2020

Expected Distances on Manifolds of Partially Oriented Flags

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.

math.DG↗

arXiv2019

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical semigroups, Wilf's conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf's conjecture, to the class of $\mathcal C$-semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf's conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by García-García, Marín-Aragón, and Vigneron-Tenorio.

math.CO↗

arXiv2019

Irreducible Generalized Numerical Semigroups and uniqueness of the Frobenius element

Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numerical semigroup is irreducible.

math.CO↗

arXiv2019

More chemical detection through less sampling: amplifying chemical signals in hyperspectral data cubes through compressive sensing

Compressive sensing (CS) is a method of sampling which permits some classes of signals to be reconstructed with high accuracy even when they were under-sampled. In this paper we explore a phenomenon in which bandwise CS sampling of a hyperspectral data cube followed by reconstruction can actually result in amplification of chemical signals contained in the cube. Perhaps most surprisingly, chemical signal amplification generally seems to increase as the level of sampling decreases. In some examples, the chemical signal is significantly stronger in a data cube reconstructed from 10% CS sampling than it is in the raw, 100% sampled data cube. We explore this phenomenon in two real-world datasets including the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset and the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset. Each of these datasets contains the release of a chemical simulant, such as glacial acetic acid, triethyl phospate, and sulfur hexafluoride, and in all cases we use the adaptive coherence estimator (ACE) to detect a target signal in the hyperspectral data cube. We end the paper by suggesting some theoretical justifications for why chemical signals would be amplified in CS sampled and reconstructed hyperspectral data cubes and discuss some practical implications.

eess.IV↗

arXiv2019

Total variation vs L1 regularization: a comparison of compressive sensing optimization methods for chemical detection

One of the fundamental assumptions of compressive sensing (CS) is that a signal can be reconstructed from a small number of samples by solving an optimization problem with the appropriate regularization term. Two standard regularization terms are the L1 norm and the total variation (TV) norm. We present a comparison of CS reconstruction results based on these two approaches in the context of chemical detection, and we demonstrate that optimization based on the L1 norm outperforms optimization based on the TV norm. Our comparison is driven by CS sampling, reconstruction, and chemical detection in two real-world datasets: the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset and the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset. Both datasets contain the release of a chemical simulant such as glacial acetic acid, triethyl phosphate, and sulfur hexafluoride. For chemical detection we use the adaptive coherence estimator (ACE) and bulk coherence, and we propose algorithmic ACE thresholds to define the presence or absence of a chemical of interest in both un-compressed data cubes and reconstructed data cubes. The un-compressed data cubes provide an approximate ground truth. We demonstrate that optimization based on either the L1 norm or TV norm results in successful chemical detection at a compression rate of 90%, but we show that L1 optimization is preferable. We present quantitative comparisons of chemical detection on reconstructions from the two methods, with an emphasis on the number of pixels with an ACE value above the threshold.

eess.IV↗

arXiv2019

A data-driven approach to sampling matrix selection for compressive sensing

Sampling is a fundamental aspect of any implementation of compressive sensing. Typically, the choice of sampling method is guided by the reconstruction basis. However, this approach can be problematic with respect to certain hardware constraints and is not responsive to domain-specific context. We propose a method for defining an order for a sampling basis that is optimal with respect to capturing variance in data, thus allowing for meaningful sensing at any desired level of compression. We focus on the Walsh-Hadamard sampling basis for its relevance to hardware constraints, but our approach applies to any sampling basis of interest. We illustrate the effectiveness of our method on the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset, the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset, and a Colorado State University Swiss Ranger depth image dataset. The spectral datasets consist of simulant experiments, including releases of chemicals such as GAA and SF6. We combine our sampling and reconstruction with the adaptive coherence estimator (ACE) and bulk coherence for chemical detection and we incorporate an algorithmic threshold for ACE values to determine the presence or absence of a chemical. We compare results across sampling methods in this context. We have successful chemical detection at a compression rate of 90%. For all three datasets, we compare our sampling approach to standard orderings of sampling basis such as random, sequency, and an analog of sequency that we term `frequency.' In one instance, the peak signal to noise ratio was improved by over 30% across a test set of depth images.

eess.SP↗