arXiv subjects

Joseph W. Iverson

Publications and source records attributed to Joseph W. Iverson.

At least 19 recordsLinked to original sources

arXiv2026

Approximation theorems in bilipschitz invariant theory

Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.

math.FA↗

arXiv2026

Totally symmetric Grassmannian codes

We introduce a general technique to construct tight fusion frames with prescribed symmetries. Applying this technique with a prescription for "all the symmetries", we construct a new family of equi-isoclinic tight fusion frames (EITFFs), which consequently form optimal Grassmannian codes. By virtue of their construction, our EITFFs have the remarkable property of total symmetry: any permutation of subspaces can be achieved by an appropriate unitary.

math.CO↗

arXiv2025

Dimension-counting bounds for equi-isoclinic subspaces

We make four contributions to the theory of optimal subspace packings and equi-isoclinic subspaces: (1) a new lower bound for block coherence, (2) an exact count of equi-isoclinic subspaces of even dimension $r$ in $\mathbb{R}^{2r+1}$ with parameter $α\neq \tfrac{1}{2}$, (3) a new upper bound for the number of $r$-dimensional equi-isoclinic subspaces in $\mathbb{R}^d$ or $\mathbb{C}^d$, and (4) a proof that when $d=2r$, a further refinement of this bound is attained for every $r$ in the complex case and every $r=2^k$ in the real case. For each of these contributions, the proof ultimately relies on a dimension count.

cs.IT↗

arXiv2025

Asymmetric SICs over finite fields

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.

math.MG↗

arXiv2025

Radon-Hurwitz Grassmannian codes

Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal code in a Grassmannian, consisting of subspaces of a finite-dimensional Hilbert space for which the smallest principal angle between any pair of them is as large as possible. EITFFs yield dictionaries with minimal block coherence and so are ideal for certain types of compressed sensing. By refining classical work of Lemmens and Seidel based on Radon-Hurwitz theory, we fully characterize EITFFs in the special case where the dimension of the subspaces is exactly one-half of that of the ambient space. We moreover show that each such "Radon-Hurwitz EITFF" is highly symmetric, where every even permutation is an automorphism.

cs.IT↗

arXiv2024

More on the optimal arrangement of $2d$ lines in $\mathbb{C}^d$

We introduce a new infinite family of $d\times 2d$ equiangular tight frames. Many matrices in this family consist of two $d\times d$ circulant blocks. We conjecture that such equiangular tight frames exist for every $d$. We show that our conjecture holds for $d\leq 165$ by a computer-assisted application of a Newton-Kantorovich theorem. In addition, we supply numerical constructions that corroborate our conjecture for $d\leq 1500$.

math.MG↗

arXiv2024

Towards a bilipschitz invariant theory

Consider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

math.FA↗

arXiv2023

On the optimal arrangement of $2d$ lines in $\mathbb{C}^d$

We show the optimal coherence of $2d$ lines in $\mathbb{C}^{d}$ is given by the Welch bound whenever a skew Hadamard of order $d+1$ exists. Our proof uses a variant of Hadamard doubling that converts any equiangular tight frame of size $\tfrac{d-1}{2} \times d$ into another one of size $d \times 2d$. Among $d < 150$, this produces equiangular tight frames of new sizes when $d = 11$, $35$, $39$, $43$, $47$, $59$, $67$, $71$, $83$, $95$, $103$, $107$, $111$, $119$, $123$, $127$, $131$, and $143$.

math.MG↗

arXiv2022

Doubly transitive lines II: Almost simple symmetries

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe involving Schur covers to recover doubly transitive lines from their automorphism group. Combining our results with recent work on the affine case by Dempwolff and Kantor, we deduce a classification of all linearly dependent doubly transitive lines in real or complex space.

math.CO↗

arXiv2022

Equi-isoclinic subspaces, covers of the complete graph, and complex conference matrices

In 1992, Godsil and Hensel published a ground-breaking study of distance-regular antipodal covers of the complete graph that, among other things, introduced an important connection with equi-isoclinic subspaces. This connection seems to have been overlooked, as many of its immediate consequences have never been detailed in the literature. To correct this situation, we first describe how Godsil and Hensel's machine uses representation theory to construct equi-isoclinic tight fusion frames. Applying this machine to Mathon's construction produces $q+1$ planes in $\mathbb{R}^{q+1}$ for any even prime power $q>2$. Despite being an application of the 30-year-old Godsil-Hensel result, infinitely many of these parameters have never been enunciated in the literature. Following ideas from Et-Taoui, we then investigate a fruitful interplay with complex symmetric conference matrices.

math.CO↗

arXiv2022

Group-invariant max filtering

Given a real inner product space $V$ and a group $G$ of linear isometries, we construct a family of $G$-invariant real-valued functions on $V$ that we call max filters. In the case where $V=\mathbb{R}^d$ and $G$ is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $V=L^2(\mathbb{R}^d)$ and $G$ is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.

cs.IT↗

arXiv2021

Harmonic Grassmannian codes

An equi-isoclinic tight fusion frame (EITFF) is a type of Grassmannian code, being a sequence of subspaces of a finite-dimensional Hilbert space of a given dimension with the property that the smallest spectral distance between any pair of them is as large as possible. EITFFs arise in compressed sensing, yielding dictionaries with minimal block coherence. Their existence remains poorly characterized. Most known EITFFs have parameters that match those of one that arose from an equiangular tight frame (ETF) in a rudimentary, direct-sum-based way. In this paper, we construct new infinite families of non-"tensor-sized" EITFFs in a way that generalizes the one previously known infinite family of them as well as the celebrated equivalence between harmonic ETFs and difference sets for finite abelian groups. In particular, we construct EITFFs consisting of $Q$ planes in $\mathbb{C}^Q$ for each prime power $Q\geq 4$, of $Q-1$ planes in $\mathbb{C}^Q$ for each odd prime power $Q$, and of $11$ three-dimensional subspaces in $\mathbb{R}^{11}$. The key idea is that every harmonic EITFF -- one that is the orbit of a single subspace under the action of a unitary representation of a finite abelian group -- arises from a smaller tight fusion frame with a nicely behaved "Fourier transform." Our particular constructions of harmonic EITFFs exploit the properties of Gauss sums over finite fields.

cs.IT↗

arXiv2021

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5.

math.CO↗

arXiv2021

Frames over finite fields: Basic theory and equiangular lines in unitary geometry

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $\mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

math.MG↗

arXiv2021

Grassmannian codes from paired difference sets

An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is "paired" with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs.

math.FA↗

arXiv2021

Doubly transitive lines I: Higman pairs and roux

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian distance-regular antipodal covers of the complete graph.

math.CO↗

arXiv2021

A note on tight projective 2-designs

We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.

cs.IT↗

arXiv2019

Multiplication-invariant operators and the classification of LCA group frames

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generated by multiplications in $L^2(X;\mathcal H)$. This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. Finally, we show applications of our results in the study of abelian group frames and translation-invariant (TI) operators acting on subspaces of $L^2(\mathcal G)$, where $\mathcal G$ is a locally compact group.

math.FA↗