arXiv subjects

Michael Bleher

Publications and source records attributed to Michael Bleher.

12 recordsLinked to original sources

arXiv2026

The Tangled Web They Weave: Exploring Neural Representations with Directed Simplicial Complexes

Understanding how artificial neural networks (ANNs) represent and combine features internally is a central challenge in mechanistic interpretability. Existing neural circuit-discovery methods represent neuron interactions as pairwise edges in a graph. But distributed neural representations consist of large groups of neurons that are inherently linked by higher-order causal dependencies: suppressing a subgroup of neurons in concert may shift the activation of other neurons in the feature representation. We propose a method to construct filtered directed simplicial complexes (DSCs) that give a topological representation of such dependencies, built from the activation profiles of an ANN on a given set of inputs. Our approach treats ANNs as structural causal models and uses counterfactual ablations to test for causal links between groups of internal neurons within specific input contexts. As a proof of concept, we apply the approach to a pair of small multilayer perceptron (MLP) architectures in a synthetic two-dimensional quadrant classification task. In these examples, DSCs recover the designed feature pathways and coordinated class transitions, and the complexes of trained networks are higher-dimensional than those of their handcrafted references. As validation, we compare our method with graph-based circuit-discovery baselines and find that their 1-skeletons agree within the relations that a weight graph can represent. These results support DSCs as context-dependent descriptions of multi-neuron computation and may thus provide a novel, orthogonal framework to study the phenomenon of distributed feature representations and polysemanticity.

cs.LG↗

arXiv2026

Distributions of metric $n$-point invariants and applications to molecular evolution

Datasets often arise as samples from a metric measure space whose geometry reflects the data-generating process. We introduce metric $n$-point invariants, defined as functions of the pairwise distances of $n$-point configurations, to study the geometry of such spaces. This provides a unified framework for existing concepts, such as the pairwise distance itself, the Gromov product, and Vietoris-Rips persistence diagrams. The distribution of a metric $n$-point invariant $F$ over random $n$-point configurations can be used to extract geometric information from the space and is obtained as the pushforward of Gromov's $n$-th curvature measure $μ_n$ along $F$. Our main result is an explicit bound for the expected $p$-Wasserstein distance between $μ_n$ and the empirical curvature measure $\hatμ_{n}$ of finitely many sampled $n$-point configurations, and carries over to the associated distributions of Lipschitz metric $n$-point invariants. From this we derive the sample size needed to achieve a prescribed accuracy and level of confidence in the approximation of the curvature measure. As an application, we use metric $n$-point invariants to study molecular evolution. We describe how site-wise nucleotide frequencies shape the corresponding distributions in Jukes-Cantor and Halpern-Bruno substitution models, and analyse five million SARS-CoV-2 genomes, revealing increasingly common failures of the four-point condition, constrained nucleotide accessibility consistent with the biased mutation spectrum, and rare cycles in persistent homology reflecting recurrent mutations across lineages. These findings demonstrate the potential of our sampling-based approach to the geometry of data as a scalable, phylogeny-free tool for detecting and quantifying evolutionary structure in large genomic datasets.

math.MG↗

arXiv2026

A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology

This article provides a review of the gauge-theoretic approach to Khovanov homology, framed in terms of a generalisation of Witten's original proposal. Concretely, the physical arguments underlying Witten's insights suggest that there is a one-parameter family of Haydys-Witten instanton Floer homology groups $HF_θ\bigl(W^4\bigr)$ for four-manifolds. At the heart of the proposal is a systematic investigation of the dimensional reductions of the Haydys-Witten equations. It is shown that on the five-dimensional cylinder $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cosθ\partial_s+\sinθw$, the Haydys-Witten equations provide flow equations for the $θ$-Kapustin-Witten equations on $W^4$. Similar reductions to lower dimensions include the twisted extended Bogomolny equations on three-manifolds and the twisted octonionic Nahm equations on one-manifolds, whose solutions provide natural boundary conditions along the boundary and corners of $W^4$. These reductions determine the indicial roots of the Haydys-Witten and $θ$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $HF_θ\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $θ$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{π/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$.

math-ph↗

arXiv2026

Adiabatic Solutions of the Haydys-Witten Equations and Symplectic Khovanov Homology

An influential conjecture by Witten states that there is an instanton Floer homology of four-manifolds with corners that in certain situations is isomorphic to Khovanov homology of a given knot $K$. The Floer chain complex is generated by Nahm pole solutions of the Kapustin-Witten equations on $\mathbb{R}^3 \times \mathbb{R}^+_y$ with an additional monopole-like singular behaviour along the knot $K$ inside the three-dimensional boundary at $y=0$. The Floer differential is given by counting solutions of the Haydys-Witten equations that interpolate between Kapustin-Witten solutions along an additional flow direction $\mathbb{R}_s$. This article investigates solutions of a decoupled version of the Haydys-Witten equations on $\mathbb{R}_s \times \mathbb{R}^3 \times \mathbb{R}^+_y$, which in contrast to the full equations exhibit a Hermitian Yang-Mills structure and can be viewed as a lift of the extended Bogomolny equations (EBE) from three to five dimensions. Inspired by Gaiotto-Witten's approach of adiabatically braiding EBE-solutions to obtain generators of the Floer homology, we propose that there is an equivalence between adiabatic solutions of the decoupled Haydys-Witten equations and non-vertical paths in the moduli space of EBE-solutions fibered over the space of monopole positions. Moreover, we argue that the Grothendieck-Springer resolution of the Lie algebra of the gauge group provides a finite-dimensional model of this moduli space of monopole solutions. These considerations suggest an intriguing similarity between Haydys-Witten instanton Floer homology and symplectic Khovanov homology and provide a novel approach towards a proof of Witten's gauge-theoretic interpretations of Khovanov homology.

math-ph↗

arXiv2026

Ultrafast topological data analysis reveals pandemic-scale dynamics of convergent evolution

Genome variants which re-occur independently across evolutionary lineages are key molecular signatures of adaptation. Inferring the dynamics of such genetic changes from pandemic-scale genomic datasets is now possible, which opens up unprecedented insight into evolutionary processes. However, existing approaches depend on the construction of accurate phylogenetic trees, which remains challenging at scale. Here we present EVOtRec, an organism-agnostic, fast and scalable Topological Data Analysis approach that enables the inference of convergently evolving genomic variants over time directly from topological patterns in the dataset, without requiring the construction of a phylogenetic tree. Using data from both simulations and published experiments, we show that EVOtRec can robustly identify variants under positive selection and performs orders of magnitude faster than state-of-the-art phylogeny-based approaches, with comparable results. We apply EVOtRec to three large viral genome datasets: SARS-CoV-2, influenza virus A subtype H5N1 and HIV-1. We identify key convergent genome variants and demonstrate how EVOtRec facilitates the real-time tracking of high fitness variants in large datasets with millions of genomes, including effects modulated by varying genomic backgrounds. We envision our Topological Data Analysis approach as a new framework for efficient comparative genomics.

q-bio.PE↗

arXiv2026

A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies

We study a family of non-local isoperimetric energies $E_{γ,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 Δ$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_Ω$ associated to a measurable set $Ω\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $Ω$ has finite perimeter if and only if $c_Ω$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $Ω$. We show that on the round sphere $E_{γ,\varepsilon}$ admits a reformulation in terms of $c_Ω$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $Γ$-convergence.

math.AP↗

arXiv2025

Low-dimensional embeddings of high-dimensional data

Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the demand for algorithms that create low-dimensional representations, or embeddings, for data visualization, exploration, and analysis is now greater than ever. In recent years, numerous embedding algorithms have been developed, and their usage has become widespread in research and industry. This surge of interest has resulted in a large and fragmented research field that faces technical challenges alongside fundamental debates, and it has left practitioners without clear guidance on how to effectively employ existing methods. Aiming to increase coherence and facilitate future work, in this review we provide a detailed and critical overview of recent developments, derive a list of best practices for creating and using low-dimensional embeddings, evaluate popular approaches on a variety of datasets, and discuss the remaining challenges and open problems in the field.

cs.LG↗

arXiv2024

An Algebraic Framework for the Modeling of Limit Order Books

Introducing an algebraic framework for modeling limit order books (LOBs) with tools from physics and stochastic processes, our proposed framework captures the creation and annihilation of orders, order matching, and the time evolution of the LOB state. It also enables compositional settings, accommodating the interaction of heterogeneous traders and different market structures. We employ Dirac notation and generalized generating functions to describe the state space and dynamics of LOBs. The utility of this framework is shown through simulations of simplified market scenarios, illustrating how variations in trader behavior impact key market observables such as spread, return volatility, and liquidity. The algebraic representation allows for exact simulations using the Gillespie algorithm, providing a robust tool for exploring the implications of market design and policy changes on LOB dynamics. Future research can expand this framework to incorporate more complex order types, adaptive event rates, and multi-asset trading environments, offering deeper insights into market microstructure and trader behavior and estimation of key drivers for market microstructure dynamics.

q-fin.TR↗

arXiv2023

The Decoupled Haydys-Witten Equations and a Weitzenböck Formula

The Haydys-Witten equations are partial differential equations on five-dimensional Riemannian manifolds that are equipped with a non-vanishing vector field $v$. Conjecturally, their solutions determine the Floer differential in a gauge-theoretic approach to Khovanov homology. This article introduces a certain decoupled version of the Haydys-Witten equations, a specialization of the Haydys-Witten equations that exhibits a Hermitian Yang-Mills structure. These equations exist whenever the vector bundle defined by the orthogonal complement of $v$ admits an almost Hermitian structure. We investigate the relation between the full Haydys-Witten equations and their decoupled version on manifolds with poly-cylindrical ends and boundaries, and find conditions under which the Haydys-Witten equations reduce to the decoupled equations. This relies on a Weitzenböck-like formula that shows that the difference between the full Haydys-Witten equations and the decoupled equations is governed by the asymptotic behaviour of solutions near boundaries and non-compact ends. Regarding the analysis near boundaries, we provide a detailed analysis of the polyhomogeneous expansion of Haydys-Witten solutions with twisted Nahm pole boundary conditions, generalizing work of Siqi He in the untwisted case. The corresponding analysis at non-compact ends relies on a vanishing theorem by Nagy-Oliveira that was recently generalized by the author.

math.DG↗

arXiv2023

Growth of the Higgs Field for Kapustin-Witten solutions on ALE and ALF gravitational instantons

The $θ$-Kapustin-Witten equations are a family of equations for a connection $A$ on a principal $G$-bundle $E \to W^4$ and a one-form $ϕ$, called the Higgs field, with values in the adjoint bundle $\operatorname{ad} E$. They give rise to second-order partial differential equations that can be studied more generally on Riemannian manifolds $W^n$ of dimension $n$. For $G=SU(2)$, we report a dichotomy that is satisfied by solutions of the second-order equations on Ricci-flat ALX spaces with sectional curvature bounded from below. This dichotomy was originally established by Taubes for $W^n=\mathbb{R}^n$; the alternatives are: either the asymptotic growth of the averaged norm of the Higgs field over geodesic spheres is larger than a positive power of the radius, or the commutator $[ϕ\wedgeϕ]$ vanishes everywhere. As a consequence, we are able to confirm a conjecture by Nagy and Oliveira, namely, that finite energy solutions of the $θ$-Kapustin-Witten equations on ALE and ALF gravitational instantons with $θ\neq 0$ are such that $[ϕ\wedgeϕ]=0$, $\nabla^A ϕ=0$, and $A$ is flat.

math.DG↗

arXiv2022

MuRiT: Efficient Computation of Pathwise Persistence Barcodes in Multi-Filtered Flag Complexes via Vietoris-Rips Transformations

Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a Vietoris-Rips transformation, a method that reduces the computation of the one-parameter persistent homology of pathwise subcomplexes in multi-filtered flag complexes to the computation of the Vietoris-Rips persistent homology of certain semimetric spaces. The corresponding pathwise persistence barcodes track persistence features of the ambient multi-filtered complex and can in particular be used to recover the rank invariant in multi-parameter persistent homology. We present MuRiT, a scalable algorithm that computes the pathwise persistence barcodes of multi-filtered flag complexes by means of Vietoris-Rips transformations. Moreover, we provide an efficient software implementation of the MuRiT algorithm which resorts to Ripser for the actual computation of Vietoris-Rips persistence barcodes. To demonstrate the applicability of MuRiT to real-world datasets, we establish MuRiT as part of our CoVtRec pipeline for the surveillance of the convergent evolution of the coronavirus SARS-CoV-2 in the current COVID-19 pandemic.

math.AT↗

arXiv2021

From orders to prices: A stochastic description of the limit order book to forecast intraday returns

We propose a microscopic model to describe the dynamics of the fundamental events in the limit order book (LOB): order arrivals and cancellations. It is based on an operator algebra for individual orders and describes their effect on the LOB. The model inputs are arrival and cancellation rate distributions that emerge from individual behavior of traders, and we show how prices and liquidity arise from the LOB dynamics. In a simulation study we illustrate how the model works and highlight its sensitivity with respect to assumptions regarding the collective behavior of market participants. Empirically, we test the model on a LOB snapshot of XETRA, estimate several linearized model specifications, and conduct in- and out-of-sample forecasts.The in-sample results based on contemporaneous information suggest that our model describes returns very well, resulting in an adjusted $R^2$ of roughly 80%. In the more realistic setting where only past information enters the model, we observe an adjusted $R^2$ around 15%. The direction of the next return can be predicted (out-of-sample) with an accuracy above 75% for time horizons below 10 minutes. On average, we obtain an RMSPE that is 10 times lower than values documented in the literature.

q-fin.TR↗