arXiv subjects

Mahya Ghandehari

Publications and source records attributed to Mahya Ghandehari.

At least 19 recordsLinked to original sources

arXiv2026

Unified Fourier transform on graphs sampled from stochastic block models

Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we show how a Fourier basis can be easily calculated from the block sizes and the block probability matrix. Using perturbation theory, we derive bounds on the sensitivity of the basis with respect to variations in the block sizes. We then consider SBMs constructed from weighted Cayley graphs. When block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. When block sizes are nearly uniform, we demonstrate that this Fourier basis closely approximates the SBM Fourier basis. For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.

eess.SP↗

arXiv2026

On amenability constants of Fourier algebras: new bounds and new examples

Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for ${\rm AM}({\rm A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where ${\rm AM}({\rm A}(G))$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with ``degenerate'' cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality.

math.FA↗

arXiv2025

Consistent sampling of Paley-Wiener functions on graphons

We study sampling methods for Paley-Wiener functions on graphons, thereby adapting and generalizing methods initially developed for graphs to the graphon setting. We then derive conditions under which such a sampling estimate is consistent with graphon convergence.

eess.SP↗

arXiv2024

On the restriction maps of the Fourier and Fourier-Stieltjes algebras over locally compact groupoids

The Fourier and Fourier-Stieltjes algebras over locally compact groupoids have been defined in a way that parallels their construction for groups. In this article, we extend the results on surjectivity or lack of surjectivity of the restriction map on the Fourier and Fourier-Stieltjes algebras of groups to the groupoid setting. In particular, we consider the maps that restrict the domain of these functions in the Fourier or Fourier-Stieltjes algebra of a groupoid to an isotropy subgroup. These maps are continuous contractive algebra homomorphisms. When the groupoid is étale, we show that the restriction map on the Fourier algebra is surjective. The restriction map on the Fourier-Stieltjes algebra is not surjective in general. We prove that for a transitive groupoid with a continuous section or a group bundle with discrete unit space, the restriction map on the Fourier-Stieltjes algebra is surjective. We further discuss the example of an HLS groupoid, and obtain a necessary condition for surjectivity of the restriction map in terms of property FD for groups, introduced by Lubotzky and Shalom. As a result, we present examples where the restriction map for the Fourier-Stieltjes algebra is not surjective. Finally, we use the surjectivity results to provide conditions for the lack of certain Banach algebraic properties, including the (weak) amenability and existence of a bounded approximate identity, in the Fourier algebra of étale groupoids.

math.FA↗

arXiv2024

Frames for signal processing on Cayley graphs

The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph's symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.

cs.IT↗

arXiv2024

Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth

The AMNM property for commutative Banach algebras is a form of Ulam stability for multiplicative linear functionals. We show that on any semilattice of infinite breadth, one may construct a weight for which the resulting weighted convolution algebra fails to have the AMNM property. Our work is the culmination of a trilogy started in [Semigroup Forum 102 (2021), no. 1, 86-103] and continued in [European J. Combin. 94 (2021), article 103311]. In particular, we obtain a refinement of the main result of the second paper, by establishing a dichotomy for union-closed set systems that has a Ramsey-theoretic flavour.

math.FA↗

arXiv2024

Robust Recovery of Robinson Property in $L^p$-Graphons: A Cut-Norm Approach

This paper investigates the Robinson graphon completion/recovery problem within the class of $L^p$-graphons, focusing on the range $5 5$, any $L^p$-graphon $w$ can be approximated by a Robinson graphon, with error of the approximation bounded in terms of $Λ(w)$. When viewing $w$ as a noisy version of a Robinson graphon, our method provides a concrete recipe for recovering a cut-norm approximation of a noiseless $w$. Given that any symmetric matrix is a special type of graphon, our results can be applicable to symmetric matrices of any size. Our work extends and improves previous results, where a similar question for the special case of $L^\infty$-graphons was answered.

math.CO↗

arXiv2024

The Large Deviation Principle for $W$-random spectral measures

The $W$-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for $W$-random graphs from [9], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by the large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the underlying graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of the integral operators corresponding to certain random graph sequences. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces of the integral operators are affected by large deviations. Potential implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing are indicated.

math.PR↗

arXiv2024

Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets

We consider the problem of characterizing the Sobolev wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, with the latter being defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive necessary and sufficient criteria for elements of the Sobolev wavefront set, formulated in terms of the decay behaviour of a given generalized continuous wavelet transform. It turns out that the characterization of directed smoothness of finite order can be performed in the two important cases: (1) bandlimited wavelets, and (2) wavelets with finitely many vanishing moments (e.g.~compactly supported wavelets). The main results of this paper are based on a number of fairly technical conditions on the dilation group. In order to demonstrate their wide applicability, we exhibit a large class of generalized shearlet groups in arbitrary dimensions fulfilling all required conditions, and give estimates of the involved constants.

math.FA↗

arXiv2024

Representing topological full groups in Steinberg algebras and C*-algebras

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.

math.OA↗

arXiv2023

Graph sequences sampled from Robinson graphons

The function $Γ$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function $Γ$ is compatible with the cut-norm $\|\cdot \|_\Box$, in the sense that graphons close in cut-norm have similar $Γ$-values. Here we show the converse, by proving that every graphon $w$ can be approximated by a Robinson graphon $R_w$ so that $\|w-R_w\|_\Box$ is bounded in terms of $Γ(w)$. We then use classical techniques from functional analysis to show that a converging graph sequence $\{G_n\}$ converges to a Robinson graphon if and only if $Γ(G_n)\rightarrow 0$. Finally, using probabilistic techniques we show that the rate of convergence of $Γ$ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly $w$ exhibits the Robinson property.

math.CO↗

arXiv2023

Signal processing on large networks with group symmetries

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is subject to error or natural changes over time. This motivated a new perspective on GSP, where the signal processing framework is developed for an entire class of graphs with similar structures. This approach can be formalized via the theory of graph limits, where graphs are considered as random samples from a distribution represented by a graphon. When the network under consideration has underlying symmetries, they may be modeled as samples from Cayley graphons. In Cayley graphons, vertices are sampled from a group, and the link probability between two vertices is determined by a function of the two corresponding group elements. Infinite groups such as the 1-dimensional torus can be used to model networks with an underlying spatial reality. Cayley graphons on finite groups give rise to a Stochastic Block Model, where the link probabilities between blocks form a (edge-weighted) Cayley graph. This manuscript summarizes some work on graph signal processing on large networks, in particular samples of Cayley graphons.

eess.SP↗

arXiv2022

A noncommutative approach to the graphon Fourier transform

Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon. Theoretically, graphons are limit objects of converging sequences of graphs. Our work extends previous research aiming to provide a common scheme for signal analysis of graphs that are similar in structure to a graphon. We extend a previous definition of graphon Fourier transform, and show that the graph Fourier transforms of graphs in a converging graph sequence converge to the graphon Fourier transform of the limiting graphon. We then apply this convergence result to signal processing on Cayley graphons. We show that Fourier analysis of the underlying group enables the construction of a suitable eigen-decomposition for the graphon, which can be used as a common framework for signal processing on graphs converging to the graphon.

math.CO↗

arXiv2022

Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach

An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked data analysis, as its vertices represent permutations, while the generating set formalizes a notion of distance between rankings. Taking advantage of the rich theory of representations of the symmetric group, we study a particular class of frames, called Frobenius-Schur frames, where every atom belongs to the coefficient space of only one irreducible representation of the symmetric group. We provide a characterization for all Frobenius-Schur frames on the group algebra of the symmetric group which are "compatible" with respect to the generating set. Such frames have been previously studied for the permutahedron, the Cayley graph of the symmetric group with the generating set of adjacent transpositions, and have proved to be capable of producing meaningful interpretation of the ranked data set via the analysis coefficients. Our results generalize frame constructions for the permutahedron to any inverse-closed generating set.

eess.SP↗

arXiv2021

Beurling-Fourier algebras on Lie groups and their spectra

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.

math.FA↗

arXiv2021

Unavoidable subprojections in union-closed set systems of infinite breadth

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, ${\mathcal T}_{\rm min}(E)$ and ${\mathcal T}_{\rm ort}(E)$. We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.

math.CO↗

arXiv2021

Dual convolution for the affine group of the real line

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $L^2({\mathbb R}^\times, dt/ |t|)$. In this paper we study the "dual convolution product" of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $L^p({\mathbb R}^\times, dt/ |t|)$ for $p\in (1,2)\cup(2,\infty)$.

math.FA↗

arXiv2021

Gabor-type frames for signal processing on graphs

In the past decade, significant progress has been made to generalize classical tools from Fourier analysis to analyze and process signals defined on networks. In this paper, we propose a new framework for constructing Gabor-type frames for signals on graphs. Our approach uses general and flexible families of linear operators acting as translations. Compared to previous work in the literature, our methods yield the sharp bounds for the associated frames, in a broad setting that generalizes several existing constructions. We also examine how Gabor-type frames behave for signals defined on Cayley graphs by exploiting the representation theory of the underlying group. We explore how natural classes of translations can be constructed for Cayley graphs, and how the choice of an eigenbasis can significantly impact the properties of the resulting translation operators and frames on the graph.

math.FA↗