arXiv:2608. 14215v1 Announce Type: new Abstract: Constrained optimization extends classical optimization by integrating side information, making it widely applicable across scientific and engineering domains.
arXiv:2609. 30477v1 Announce Type: cross Abstract: Exact Euclidean \(K\)-means partitions \(n\) observations into \(K\) unlabelled clusters, but the unrestricted search is generally exponential.
arXiv:2411. 12438v2 Announce Type: replace-cross Abstract: We develop a new approach for clustering non-spherical (i.
The paper formalizes a geometric tradeoff between ambient separation and sampling gaps to determine when distinct manifold components can be reliably separated in clustering. It introduces a threshold phenomenon for mutual‑k‑nearest‑neighbor graphs, defining an uncertainty zone where the number of clusters cannot be identified. The authors propose Manifold‑Based Clustering (MBC), which outputs a bracket interval quantifying this uncertainty rather than forcing a single cluster count.
arXiv:2502. 08397v3 Announce Type: replace-cross Abstract: Clustering is a fundamental technique in data analysis and machine learning, used to group similar data points together.
arXiv:2607. 09490v1 Announce Type: cross Abstract: Terminal embeddings have emerged as a powerful tool for dimension reduction.
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:2607. 24237v1 Announce Type: new Abstract: Many existing clustering methods are designed based on a set-oriented definition---a cluster is a set of similar points---relying a point-to-point similarity function to find similar points.
arXiv:2602. 08542v3 Announce Type: replace-cross Abstract: Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center.
arXiv:2609.06394v1 Announce Type: cross Abstract: Massive datasets in modern machine learning have made data reduction a central challenge, particularly for clustering tasks where memory and computat...
arXiv:2606. 29261v1 Announce Type: new Abstract: We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables.
arXiv:2607. 24537v1 Announce Type: new Abstract: In the Big Data era, the scalability of clustering algorithms constitutes a key challenge.
arXiv:2609.36762v1 Announce Type: new Abstract: Federated clustering methods that do not require the global number of clusters $K$ still assume that each client knows its local number $K_g$. This ass...