arXiv subjects

Alexander Van Werde

Publications and source records attributed to Alexander Van Werde.

13 recordsLinked to original sources

arXiv2026

Spectral graph clustering with inhomogeneous latent geometry

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

cs.SI↗

arXiv2026

Finding all cospectral mates over a number field

We investigate a notion of cospectrality for integer matrices that is parameterized by algebraic number fields. Given a number field and a symmetric integer matrix, we wonder when conjugating the integer matrix by an orthogonal matrix with entries in the given field can produce new integer matrices. Our results concern sufficient conditions for the associated notion of spectral determination, and we give constraints on the orthogonal matrices when the conditions are not applicable. The results use the discriminant of the characteristic polynomial and properties of Krylov subspaces. We leverage the theory to develop an algorithm to find all cospectral mates over a given (small) field. An implementation of the algorithm is made available.

math.NT↗

arXiv2026

Detection and Evaluation of Clusters within Sequential Data

Sequential data is ubiquitous -- it is routinely gathered to gain insights into complex processes such as behavioral, biological, or physical processes. Challengingly, such data not only has dependencies within the observed sequences, but the observations are also often high-dimensional, sparse, and noisy. These are all difficulties that obscure the inner workings of the complex process under study. One solution is to calculate a low-dimensional representation that describes (characteristics of) the complex process. This representation can then serve as a proxy to gain insight into the original process. However, uncovering such low-dimensional representation within sequential data is nontrivial due to the dependencies, and an algorithm specifically made for sequences is needed to guarantee estimator consistency. Fortunately, recent theoretical advancements on Block Markov Chains have resulted in new clustering algorithms that can provably do just this in synthetic sequential data. This paper presents a first field study of these new algorithms in real-world sequential data; a wide empirical study of clustering within a range of data sequences. We investigate broadly whether, when given sparse high-dimensional sequential data of real-life complex processes, useful low-dimensional representations can in fact be extracted using these algorithms. Concretely, we examine data sequences containing GPS coordinates describing animal movement, strands of human DNA, texts from English writing, and daily yields in a financial market. The low-dimensional representations we uncover are shown to not only successfully encode the sequential structure of the data, but also to enable gaining new insights into the underlying complex processes.

cs.LG↗

arXiv2026

Sharp concentration for sums of matrices with Markovian dependence through universality

We prove that a sum of random matrices generated by a $ψ$-mixing Markov chain has similar spectral properties to a Gaussian matrix with the same mean and covariance structure. This nonasymptotic universality principle enables sharp concentration inequalities when combined with recent advances in the Gaussian literature. We illustrate the theory with examples, showing how it enables polynomial dimensional improvements relative to previous Markovian matrix concentration results when applied to Wigner-type matrices, and how one can recover sharp limiting values for a model used to study spectral clustering techniques. A key challenge in the proof is that techniques based only on classical cumulants, which can be used when summands are independent, are not sufficient on their own for efficient estimates in a Markovian setting. Our approach exploits Boolean cumulants and a change--of--measure argument.

math.PR↗

arXiv2026

On the satisfaction frequency of spectral characterization conditions

We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial.

math.PR↗

arXiv2026

Are sparse graphs typically determined by their spectrum?

We investigate whether it is typical for a sparse graph to be uniquely characterized by its adjacency spectrum up to isomorphism. Our first result shows that the giant component of an Erdős-Rényi graph is cospectral when the average degree is sufficiently small. The proof relies on the existence of a specific pendant tree, combined with a method by Schwenk that swaps trees to construct a cospectral mate. It seems possible that pendant trees are essentially the only obstruction, meaning that the giant should become characterized by spectrum with high probability if one prunes these by considering the 2-core. The majority of the paper is devoted to theoretical and numerical evidence supporting this concept. Our main theorem in this direction establishes that local switching methods can not cause the 2-core to be cospectral. We also discuss R-cospectrality and rational cospectrality at fixed level.

math.CO↗

arXiv2026

A sufficient condition for generalized spectral characterization of graphs with loops

Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.

math.CO↗

arXiv2026

Exact cospectrality probabilities for uniform random matrices

We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder what the probability is that the conjugate is again integral. Our main result establishes an exact formula for this probability in terms of the Smith ideals associated to the orthogonal matrix. As an illustrative application, we establish exact formulas for the expected number of rational orthogonal matrices that preserve the integrality of a random matrix for every fixed denominator in dimensions two and three. Notably, the dependence on the denominator turns out to be non-monotone due to number-theoretic fluctuations. We also prove bounds on the probability of rational cospectrality with bounded but arbitrarily large denominator.

math.PR↗

arXiv2025

Worst-case mixing estimates for Brownian motion with semipermeable barriers

We study the mixing properties of a Brownian motion whose movements are hindered by semipermeable barriers. Our setting assumes that the process takes values in a smooth planar domain and that the barriers are one-dimensional closed curves. We establish an upper bound on the mixing time and a lower bound on the stationary distribution in terms of geometric parameters. These worst-case bounds decay at an exponential rate as the domain grows large, and we give examples that show that exponential decay is necessary in our worst-case setting.

math.PR↗

arXiv2025

Cokernel statistics for walk matrices of directed and weighted random graphs

The walk matrix associated to an $n\times n$ integer matrix $X$ and an integer vector $b$ is defined by $W := (b,X b, . . . ,X^{n-1} b)$. We study limiting laws for the cokernel of $W$ in the scenario where $X$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^{m}$-torsion part of the cokernel, as a group, when $X$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the $\mathbb{Z}[x]$-module structure. The motivation for this work arises from an open problem in spectral graph theory which asks to show that random graphs are often determined up to isomorphism by their (generalized) spectrum. Sufficient conditions for generalized spectral determinacy can namely be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper

math.CO↗

arXiv2024

Recovering semipermeable barriers from reflected Brownian motion

We study the recovery of one-dimensional semipermeable barriers for a stochastic process in a planar domain. The considered process acts like Brownian motion when away from the barriers and is reflected upon contact until a sufficient but random amount of interaction has occurred, determined by the permeability, after which it passes through. Given a sequence of samples, we wonder when one can determine the location and shape of the barriers. This paper identifies several different recovery regimes, determined by the available observation period and the time between samples, with qualitatively different behavior. The observation period $T$ dictates if the full barriers or only certain pieces can be recovered, and the sampling rate significantly influences the convergence rate as $T\to \infty$. This rate turns out polynomial for fixed-frequency data, but exponentially fast in a high-frequency regime. Further, the environment's impact on the difficulty of the problem is quantified using interpretable parameters in the recovery guarantees, and is found to also be regime-dependent. For instance, the curvature of the barriers affects the convergence rate for fixed-frequency data, but becomes irrelevant when $T\to \infty$ with high-frequency data. The results are accompanied by explicit algorithms, and we conclude by illustrating the application to real-life data.

math.PR↗

arXiv2023

Singular value distribution of dense random matrices with block Markovian dependence

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is $Θ(n^2)$ with $n$ the size of the state space. The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.

math.PR↗

arXiv2023

Estimates for zero loci of Bernstein-Sato ideals

We give estimates for the zero loci of Bernstein-Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein-Sato polynomials. The lower bounds generalise the fact that log-canonical thresholds, small jumping numbers of multiplier ideals, and their real versions provide roots of Bernstein-Sato polynomials.

math.AG↗