arXiv:2608.21466v1 Announce Type: new Abstract: We develop spectral algorithms for selecting state-space partitions that define averaging kernels for finite, ergodic and reversible Markov chains. For...
The paper investigates high‑dimensional graph combinatorial hypothesis testing where a signal resides on the edges of an unknown subgraph. For Gaussian observations, it derives sharp statistical and computational thresholds for detecting the signal across various graph alternatives, using second‑moment bounds of likelihood ratios and low‑degree methods. The authors also establish a transfer principle linking exponential family models, identify new statistical‑computational gaps for structures like multiple planted cliques and hypercliques, and provide matching upper bounds under local regularity, thereby revealing universal threshold scales and connecting Gaussian and Erdős–Rényi settings.
The paper introduces the Universal Clustering Problem (UCP), a framework that captures the optimisation core common to many clustering methods by maximizing a polynomial‑time computable partition utility over a finite metric space. It proves UCP is NP‑hard through reductions from graph colouring and exact cover by 3‑sets, showing that popular algorithms such as k‑means, GMMs, DBSCAN, spectral clustering, and affinity propagation inherit this intractability. The authors argue that this unified hardness explains typical failure modes—like local optima and greedy merge traps—and suggest moving toward stability‑aware objectives and interaction‑driven formulations with explicit guarantees.
arXiv:2610.33177v1 Announce Type: new Abstract: The Gaussian Mixture Model (abbrv. GMM) is a classic generative model for clustered data, for which spectral clustering based on dimension reduction vi...
arXiv:2401. 10927v3 Announce Type: replace-cross Abstract: In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions in $\mathbb{R}^p$.
arXiv:2608. 10845v1 Announce Type: cross Abstract: Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian.
arXiv:2608. 08704v1 Announce Type: cross Abstract: Kernel spectral clustering with a single bandwidth can be inadequate for data exhibiting multiple characteristic pairwise-distance scales, a problem particularly prevalent in the high-dimensional regime.
arXiv:2606. 19411v3 Announce Type: replace Abstract: Selecting a fixed-size subset that maximizes the determinant of a positive semidefinite kernel is the MAP problem for a size-constrained determinantal point process and the classical maximum-entropy sampling problem.
arXiv:2606. 05230v1 Announce Type: cross Abstract: Selecting a clustering algorithm and its hyperparameters without labels is a common difficulty in engineering machine learning pipelines that work with unsupervised analysis of sensor, image, or process data.
arXiv:2411.02835v3 Announce Type: replace-cross Abstract: The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral cluster...
arXiv:2607. 21039v1 Announce Type: new Abstract: Spectral methods are among the most widely used techniques for community detection, clustering, and graph learning.
arXiv:2605. 22346v2 Announce Type: replace-cross Abstract: Two of the most widely used methods for analysing graph data, Adjacency Spectral Embedding and Laplacian Spectral Embedding, often produce different results when applied to the same graph.