Spectral Clustering

Dense self-expression matrices and full-affinity spectral clustering limit the scalability of subspace clustering. We introduce the Latent Orthogonal Optimization Model for Subspace Clustering (LoomSC), a framework that addresses both bottlenecks through projector factorization and exact spectral reduction. Motivated by the spectral structure of least-squares regression, LoomSC jointly learns latent features and a projector self-representation through two thin factors. Alternating Procrustes and least-squares updates preserve the sample factor's orthogonality while keeping the coefficient matrix implicit. We construct a nonnegative quadratic affinity that preserves the projector's support. An exact feature map then reduces its normalized spectral problem to an eigenproblem whose dimension depends only on the factor width. Neither the full affinity nor the sample Laplacian needs to be formed. Our analysis quantifies the projector approximation and identifies conditions for subspace preservation and within-subspace connectivity. For fixed dimensions and iteration budgets, the complete pipeline has linear time and memory complexity in the number of samples. Across five image-clustering benchmarks, LoomSC ranks first or second in all 15 dataset-metric comparisons against 9 state-of-the-art baselines. Its mean accuracy exceeds the highest baseline mean by 6.66 percentage points. Synthetic experiments scale to 500,000 samples while maintaining at least 99.8% accuracy.

Spectral co-clustering is a useful tool for discovering latent structure in word-document matrices, but its reliance on singular value decomposition (SVD) can make standard formulations expensive on high-dimensional data. This paper presents two randomized approximations for normalized spectral co-clustering of bipartite text data when the numbers of document and word clusters may differ. The first method uses randomized SVD through random projection, while the second combines partial SVD with element-wise random sampling. Across real-world and synthetic datasets, both methods reduce runtime relative to the full-SVD baseline, but their behavior depends on matrix sparsity. The random projection method is the more reliable approximation across the tested settings, whereas the sampling-based method is most useful on denser matrices and provides limited benefit on already sparse text data. These results show that randomized approximations for spectral co-clustering should be selected according to the underlying structure of the data.

Taxonomy induction aims to organize concept sets into coherent hierarchical structures. Recent LLM-based methods can induce taxonomies directly from flat term lists, avoiding the need for corpora, but degrade sharply as concept sets scale up. We argue that this degradation stems not only from context length limitations, but also from structural failures in hierarchical reasoning. To address this, we adopt a divide-and-merge paradigm that partitions concepts into smaller subsets, induces local taxonomies, and merges them into a global hierarchy. However, we identify two structural failure modes inherent to this paradigm: Structural Fragmentation, where partitioning weakens local hierarchical signals, and Parent Displacement, where locally plausible relations are misplaced in the global hierarchy. To address both, we propose SPARROW, a scalable taxonomy induction framework that combines structure-preserving spectral partitioning to retain hierarchical connectivity within each block, and constraint-guided incremental fusion that treats block-level relations as structural constraints rather than ground truth for global placement. Experiments on large-scale benchmarks show that SPARROW consistently achieves the strongest global structural quality across backbones. The code is available at https://github.com/rebeccazyr/SPARROW.

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.

Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian. We study a continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases. Under a random dot product graph model, we establish a row-wise central limit theorem for this family of embeddings. The result provides an explicit description of how degree normalization affects both population geometry and the local uncertainty of embedded nodes. We use the limiting distributions to compare different normalizations in two-community stochastic block models through a projected-Gaussian Bayes-error diagnostic. These comparisons show that no single normalization is uniformly preferred. Instead, the favored normalization depends on network density, community imbalance, and block-probability structure. Typically, stronger normalization is favored in lower-density or more imbalanced settings. These results provide a unified distributional understanding of when and why alternative normalizations may improve spectral clustering.

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. We address this issue through a multi-kernel formulation that aggregates kernels with different bandwidths. The bandwidths are selected as prescribed empirical quantiles of the pairwise squared distances, thereby capturing the relevant distance scales without requiring prior population-scale information. We develop a rigorous theoretical analysis of the resulting method under a general high-dimensional, multi-scale mixture model with heterogeneous cluster centers and covariance geometries. We construct a blockwise constant, low-rank informative approximation to the empirical multi-kernel matrix and establish row-wise $\ell_{2,\infty}$ perturbation bounds for its leading spectral components, as well as for the associated normalized Laplacian matrix. These bounds yield observation-level control of the spectral embedding, which is more informative than conventional global eigenspace perturbation estimates. Under suitable eigen-gap and cluster-separation conditions, we show that approximate $K$-means applied to the multi-kernel spectral embedding achieves exact recovery with high probability.

This work focuses on the problem of learning on temporal graphs, with particular emphasis on the task of clustering: obtaining coarse-grained representations by aggregating information from nodes, edges, and temporal dynamics - a task related to pooling in machine learning on graphs, or community detection in network science. Although graph neural networks reach state-of-the-art performance across many downstream graph tasks, their advantage over established descriptive and inferential clustering algorithms is far less settled, especially under demands of efficiency and recovery accuracy. We frame this tension through three linked perspectives: principles, connecting graph learning and community detection through shared spectral foundations and detectability thresholds in stochastic block model regimes; primitives, making spectral clustering and multislice modularity optimization tractable through GPU-accelerated temporal backends; and pooling, viewing principled community detection as a theory-grounded coarse-graining operator for temporal graphs. Our results indicate that algorithmic methods remain the appropriate tool where attributes are absent or weak - scalability rather than accuracy being the binding obstacle - while neural models are most compelling when structural, temporal, and attribute signals align. By making temporal clustering scalable, GPU-accelerated primitives suggest a route toward theory-grounded pooling, while raising a central question: when does community-based coarse-graining preserve the dynamics needed for downstream learning tasks?

This work addresses community detection in temporal networks through GPU-accelerated extensions of spectral clustering and modularity-based algorithms originally designed for static graphs. Built on the NVIDIA RAPIDS ecosystem, the framework enables the characterization and tracking of communities in snapshot-based dynamic graphs, either by Leiden greedy optimization with multi-GPU support via Dask-based workload distribution, or eigendecomposition of a symmetric Bethe-Hessian operator. Our multislice modularity backend achieves up to roughly three orders of magnitude speedup over the CPU reference under an equal-work budget, depending on graph density and snapshot count, while preserving compatibility with existing graph analytics pipelines. We demonstrate its applicability on real-world and synthetic datasets, facilitating exploratory analysis of structural network properties over time. Such capabilities are relevant across several application domains, such as epidemic spreading, financial systems, cybersecurity, and trajectory and mobility analysis. We release our implementation as free and open-source software, including Python bindings through the NetworkX-Temporal library for ease of use and zero-code acceleration with existing codebases.

We propose \textbf{CaSPECT}, a causal spectral clustering framework for discovering causally homogeneous subgroups from observational data. Rather than clustering in covariate space, CaSPECT defines similarity through the topology of a learned directed acyclic graph (DAG); a bootstrap-stabilised PC algorithm recovers the causal skeleton; a novel \emph{Orientation Validation Score} (OVS) combines PC bootstrap evidence with DirectLiNGAM to orient edges robustly; directed edges are weighted by backdoor-identified average treatment effects estimated via OLS or double machine learning. Chung's directed Laplacian provides a spectral embedding in which individuals close together share the same causal propagation pathways. We establish almost-sure consistency of the full pipeline and validate the method through a controlled simulation study and on LaLonde CPS1, IHDP, and 401(k) datasets, where CaSPECT recovers a positive and statistically significant treatment effect within the causally comparable subpopulation and corrects for severe confounding without requiring a pre-specified propensity score model.

Spectral clustering largely depends on the affinity graph, yet constructing a graph that preserves reliable local connectivity while adapting to heterogeneous data structures remains challenging. Existing granular-ball-based spectral clustering methods usually reduce graph complexity by using coarse-grained representatives. However, the learned local regions are often treated as graph nodes or anchors, and their structural information is not sufficiently used to regularize the original sample-level graph. To address this issue, this paper proposes a Minimum Description Length based Granular-Ball Tree-Regularized Spectral Clustering method, termed MDL-GBTRSC. The proposed method constructs a granular-ball tree through local MDL model selection, with reciprocal neighborhood continuity used to discourage splits that break reliable local connections. The stable leaf balls obtained from the tree provide coding-scale information for regularizing the sample-level affinity graph. In addition, a shared-neighbor bridge code is introduced to adjust weak local bridge relations without requiring an additional user-specified threshold. In this way, MDL-GBTRSC connects interpretable local representation learning with affinity graph construction in a unified spectral clustering framework. Experiments on real and synthetic datasets show that MDL-GBTRSC achieves the best average ARI and NMI under the adopted fixed-configuration protocol compared with classical spectral clustering baselines and representative granular-ball, micro-cluster, and anchor-based methods.

Collections of interacting AI agents can form coalitions, creating emergent group-level organization that is critical for AI safety and alignment. However, observing agent behavior alone is often insufficient to distinguish genuine informational coupling from spurious similarity, as consequential coalitions may form at the level of internal representations before any overt behavioral change is apparent. Here, we introduce a practical method for detecting coalition structure from the internal neural representations of multi-agent systems. The approach constructs a pairwise mutual-information graph from the hidden states of agents and applies spectral partitioning to identify the most salient coalition boundary. We validate this method in two domains. First, in multi-agent reinforcement learning environments, the method successfully recovers programmed hierarchical and dynamic coalition structures and correctly rejects false positives arising from behavioral coordination without informational coupling. Second, using a large language model, the method identifies coalition structures implied by descriptive prompts, tracks dynamic team reassignments, and reveals a representational hierarchy where explicit labels dominate over conflicting interaction patterns. Across both settings, the recovered partition reveals subgroup organization that a scalar cross-agent mutual-information measure cannot distinguish. The results demonstrate that analyzing hidden-state mutual information through spectral partitioning provides a scalable diagnostic for identifying representational coalitions, offering a valuable tool for monitoring emergent structure in distributed AI systems.

We study the community detection problem in the non-uniform hypergraph stochastic block model (HSBM), where hyperedges of varying sizes coexist. This setting captures higher-order and multi-view interactions and raises a fundamental question: can multiple uniform hypergraph layers below the detection threshold be combined to enable weak recovery? We answer this question by establishing a Kesten--Stigum-type bound for weak recovery in a general class of non-uniform HSBMs with $r$ blocks, generated according to multiple symmetric probability tensors. In the case $r=2$, we show that weak recovery is possible whenever the sum of the signal-to-noise ratios across all uniform hypergraph layers exceeds one, thereby confirming the positive part of a conjecture in (Chodrow et al., 2023). Moreover, we provide a polynomial-time spectral algorithm that achieves this threshold via an optimally weighted non-backtracking operator. For the unweighted non-backtracking matrix, our spectral method attains a different algorithmic threshold, also conjectured in (Chodrow et al., 2023). Our approach develops a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs, including a precise characterization of outlier eigenvalues and eigenvector overlaps. We introduce a novel Ihara--Bass formula tailored to weighted non-uniform hypergraphs, which yields an efficient low-dimensional representation and leads to a provable spectral reconstruction algorithm. Taken together, these results provide a principled and computationally efficient approach to clustering in non-uniform hypergraphs, and highlight the role of optimal weighting in aggregating heterogeneous higher-order interactions.