arXiv subjects

Mehmet Emin Aktas

Publications and source records attributed to Mehmet Emin Aktas.

13 recordsLinked to original sources

arXiv2026

On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams

Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector spaces or compared through kernels. Although many persistence diagram vectorizations and persistence kernels have been proposed, there is no general framework for determining which persistence diagram vectorizations are synthesizable. Spectral synthesis asks whether the exponential monomials contained in the translation-invariant variety generated by a function span a dense subspace of that variety. We introduce uniformly discrete spectral synthesis for persistence diagram vectorizations using Fourier analysis on groups of virtual persistence diagrams. When the space of persistence intervals is separable, we prove sufficient conditions for uniformly discrete spectral synthesis and derive approximation results for virtual persistence diagrams with bounded total multiplicity. We then classify many persistence diagram vectorizations and persistence kernels from the literature by uniformly discrete spectral synthesis.

math.FA↗

arXiv2026

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

math.KT↗

arXiv2025

Topological Conditioning for Mammography Models via a Stable Wavelet-Persistence Vectorization

Breast cancer is the most commonly diagnosed cancer in women and a leading cause of cancer death worldwide. Screening mammography reduces mortality, yet interpretation still suffers from substantial false negatives and false positives, and model accuracy often degrades when deployed across scanners, modalities, and patient populations. We propose a simple conditioning signal aimed at improving external performance based on a wavelet based vectorization of persistent homology. Using topological data analysis, we summarize image structure that persists across intensity thresholds and convert this information into spatial, multi scale maps that are provably stable to small intensity perturbations. These maps are integrated into a two stage detection pipeline through input level channel concatenation. The model is trained and validated on the CBIS DDSM digitized film mammography cohort from the United States and evaluated on two independent full field digital mammography cohorts from Portugal (INbreast) and China (CMMD), with performance reported at the patient level. On INbreast, augmenting ConvNeXt Tiny with wavelet persistence channels increases patient level AUC from 0.55 to 0.75 under a limited training budget.

cs.CV↗

arXiv2025

Playing Devil's Advocate: Unmasking Toxicity and Vulnerabilities in Large Vision-Language Models

The rapid advancement of Large Vision-Language Models (LVLMs) has enhanced capabilities offering potential applications from content creation to productivity enhancement. Despite their innovative potential, LVLMs exhibit vulnerabilities, especially in generating potentially toxic or unsafe responses. Malicious actors can exploit these vulnerabilities to propagate toxic content in an automated (or semi-) manner, leveraging the susceptibility of LVLMs to deception via strategically crafted prompts without fine-tuning or compute-intensive procedures. Despite the red-teaming efforts and inherent potential risks associated with the LVLMs, exploring vulnerabilities of LVLMs remains nascent and yet to be fully addressed in a systematic manner. This study systematically examines the vulnerabilities of open-source LVLMs, including LLaVA, InstructBLIP, Fuyu, and Qwen, using adversarial prompt strategies that simulate real-world social manipulation tactics informed by social theories. Our findings show that (i) toxicity and insulting are the most prevalent behaviors, with the mean rates of 16.13% and 9.75%, respectively; (ii) Qwen-VL-Chat, LLaVA-v1.6-Vicuna-7b, and InstructBLIP-Vicuna-7b are the most vulnerable models, exhibiting toxic response rates of 21.50%, 18.30% and 17.90%, and insulting responses of 13.40%, 11.70% and 10.10%, respectively; (iii) prompting strategies incorporating dark humor and multimodal toxic prompt completion significantly elevated these vulnerabilities. Despite being fine-tuned for safety, these models still generate content with varying degrees of toxicity when prompted with adversarial inputs, highlighting the urgent need for enhanced safety mechanisms and robust guardrails in LVLM development.

cs.CR↗

arXiv2024

Topology-guided Hypergraph Transformer Network: Unveiling Structural Insights for Improved Representation

Hypergraphs, with their capacity to depict high-order relationships, have emerged as a significant extension of traditional graphs. Although Graph Neural Networks (GNNs) have remarkable performance in graph representation learning, their extension to hypergraphs encounters challenges due to their intricate structures. Furthermore, current hypergraph transformers, a special variant of GNN, utilize semantic feature-based self-attention, ignoring topological attributes of nodes and hyperedges. To address these challenges, we propose a Topology-guided Hypergraph Transformer Network (THTN). In this model, we first formulate a hypergraph from a graph while retaining its structural essence to learn higher-order relations within the graph. Then, we design a simple yet effective structural and spatial encoding module to incorporate the topological and spatial information of the nodes into their representation. Further, we present a structure-aware self-attention mechanism that discovers the important nodes and hyperedges from both semantic and structural viewpoints. By leveraging these two modules, THTN crafts an improved node representation, capturing both local and global topological expressions. Extensive experiments conducted on node classification tasks demonstrate that the performance of the proposed model consistently exceeds that of the existing approaches.

cs.LG↗

arXiv2023

Hypergraph Classification via Persistent Homology

Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including clustering, graph classification, and graph neural networks. The definition of persistent homology for graphs is relatively straightforward, as graphs possess distinct intrinsic distances and a simplicial complex structure. However, hypergraphs present a challenge in preserving topological information since they may not have a simplicial complex structure. In this paper, we define several topological characterizations of hypergraphs in defining hypergraph persistent homology to prioritize different higher-order structures within hypergraphs. We further use these persistent homology filtrations in classifying four different real-world hypergraphs and compare their performance to the state-of-the-art graph neural network models. Experimental results demonstrate that persistent homology filtrations are effective in classifying hypergraphs and outperform the baseline models. To the best of our knowledge, this study represents the first systematic attempt to tackle the hypergraph classification problem using persistent homology.

math.AT↗

arXiv2022

Liars are more influential: Effect of Deception in Influence Maximization on Social Networks

Detecting influential users, called the influence maximization problem on social networks, is an important graph mining problem with many diverse applications such as information propagation, market advertising, and rumor controlling. There are many studies in the literature for influential users detection problem in social networks. Although the current methods are successfully used in many different applications, they assume that users are honest with each other and ignore the role of deception on social networks. On the other hand, deception appears to be surprisingly common among humans within social networks. In this paper, we study the effect of deception in influence maximization on social networks. We first model deception in social networks. Then, we model the opinion dynamics on these networks taking the deception into consideration thanks to a recent opinion dynamics model via sheaf Laplacian. We then extend two influential node detection methods, namely Laplacian centrality and DFF centrality, for the sheaf Laplacian to measure the effect of deception in influence maximization. Our experimental results on synthetic and real-world networks suggest that liars are more influential than honest users in social networks.

cs.SI↗

arXiv2021

Identifying critical higher-order interactions in complex networks

Information diffusion on networks is an important concept in network science observed in many situations such as information spreading and rumor controlling in social networks, disease contagion between individuals, cascading failures in power grids. The critical interactions in networks are the ones that play critical roles in information diffusion and primarily affect network structure and functions. Besides, interactions can occur between not only two nodes as pairwise interactions, i.e., edges, but also three or more nodes, described as higher-order interactions. This report presents a novel method to identify critical higher-order interactions. We propose two new Laplacians that allow redefining classical graph centrality measures for higher-order interactions. We then compare the redefined centrality measures using the Susceptible-Infected-Recovered (SIR) simulation model. Experimental results suggest that the proposed method is promising in identifying critical higher-order interactions.

cs.SI↗

arXiv2021

Hypergraph Laplacians in Diffusion Framework

Networks are important structures used to model complex systems where interactions take place. In a basic network model, entities are represented as nodes, and interaction and relations among them are represented as edges. However, in a complex system, we cannot describe all relations as pairwise interactions, rather should describe as higher-order interactions. Hypergraphs are successfully used to model higher-order interactions in complex systems. In this paper, we present two new hypergraph Laplacians based on diffusion framework. Our Laplacians take the relations between higher-order interactions into consideration, hence can be used to model diffusion on hypergraphs not only between vertices but also higher-order structures. These Laplacians can be employed in different network mining problems on hypergraphs, such as social contagion models on hypergraphs, influence study on hypergraphs, and hypergraph classification, to list a few.

cs.SI↗

arXiv2020

Graph Classification via Heat Diffusion on Simplicial Complexes

In this paper, we study the graph classification problem in vertex-labeled graphs. Our main goal is to classify the graphs comparing their higher-order structures thanks to heat diffusion on their simplices. We first represent vertex-labeled graphs as simplex-weighted super-graphs. We then define the diffusion Frechet function over their simplices to encode the higher-order network topology and finally reach our goal by combining the function values with machine learning algorithms. Our experiments on real-world bioinformatics networks show that using diffusion Fr{é}chet function on simplices is promising in graph classification and more effective than the baseline methods. To the best of our knowledge, this paper is the first paper in the literature using heat diffusion on higher-dimensional simplices in a graph mining problem. We believe that our method can be extended to different graph mining domains, not only the graph classification problem.

cs.SI↗

arXiv2019

Persistence Homology of Networks: Methods and Applications

Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks. In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.

math.AT↗

arXiv2017

A polynomial invariant for plane curve complements: Krammer polynomials

We use the Krammer representation of the braid group in Libgober's invariant and construct a new multivariate polynomial invariant for curve complements: Krammer polynomial. We show that the Krammer polynomial of an essential braid is equal to zero. We also compute the Krammer polynomials of some certain n-gonal curves.

math.AG↗

arXiv2017

Dessins d'Enfants of Trigonal Curves

In this paper, we focus on properties of dessins d'enfants associated to trigonal curves. Degtyarev studied dessins d'enfants to compute braid monodromies and fundamental groups of trigonal curves using their combinatorial data. We first classify all possible combinatorial data that can occur for trigonal curves of low degree, as well as bounds on the number of possibilities for all degree. We also study deformations of trigonal curves and corresponding deformations of their dessins. Of special interest to Degtyarev was the case when the dessins are maximal. We give a sufficient condition for a trigonal curve to be deformable to one that is maximal.

math.AG↗