Hierarchical clustering is a powerful tool for exploratory data analysis, organizing data into a tree of clusterings from which a partition can be chosen. This paper generalizes these ideas by proving that, for any reasonable hierarchy, one can optimally solve any center-based clustering objective over it (such as k-means). Moreover, these solutions can be found exceedingly quickly and are themselves necessarily hierarchical. Thus, given a cluster tree, we show that one can quickly access a plethora of new, equally meaningful hierarchies. Just as in standard hierarchical clustering, one can then choose any desired partition from these new hierarchies. We conclude by verifying the utility of our proposed techniques across datasets, hierarchies, and partitioning schemes.

Data-driven algorithm selection is a powerful approach for choosing effective heuristics for computational problems. It operates by evaluating a set of candidate algorithms on a collection of representative training instances and selecting the one with the best empirical performance. However, running each algorithm on every training instance is computationally expensive, making scalability a central challenge. In practice, a common workaround is to evaluate algorithms on smaller proxy instances derived from the original inputs. However, this practice has remained largely ad hoc and lacked theoretical grounding. We provide the first theoretical foundations for this practice by formalizing the notion of size generalization: predicting an algorithm's performance on a large instance by evaluating it on a smaller, representative instance, subsampled from the original instance. We provide size generalization guarantees for three widely used clustering algorithms (single-linkage, k-means++, and Gonzalez's k-centers heuristic) and two canonical max-cut algorithms (Goemans-Williamson and Greedy). We characterize the subsample size sufficient to ensure that performance on the subsample reflects performance on the full instance, and our experiments support these findings.

Existing multi-view clustering methods employ various strategies to address datalevel sparsity and view-level dynamic fusion. However, we identify a critical yet overlooked issue: varying sparsity across views. Cross-view sparsity variations lead to encoding discrepancies, heightening sample-level semantic heterogeneity and making view-level dynamic weighting inappropriate. To tackle these challenges, we propose Adaptive Sparse Autoencoders for Multi-View Clustering (SparseMVC), a framework with three key modules. Initially, the sparse autoencoder probes the sparsity of each view and adaptively adjusts encoding formats via an entropymatching loss term, mitigating cross-view inconsistencies. Subsequently, the correlation-informed sample reweighting module employs attention mechanisms to assign weights by capturing correlations between early-fused global and viewspecific features, reducing encoding discrepancies and balancing contributions. Furthermore, the cross-view distribution alignment module aligns feature distributions during the late fusion stage, accommodating datasets with an arbitrary number of views. Extensive experiments demonstrate that SparseMVC achieves state-of-theart clustering performance.

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic k-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed k-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.

Decision trees are prized for their interpretability and strong performance on tabular data. Yet, their reliance on simple axis-aligned linear splits often forces deep, complex structures to capture non-linear feature effects, undermining human comprehension of the constructed tree. To address this limitation, we propose a novel generalization of a decision tree, the Shape Generalized Tree (SGT), in which each internal node applies a learnable axis-aligned shape function to a single feature, enabling rich, non-linear partitioning in one split. As users can easily visualize each node's shape function, SGTs are inherently interpretable and provide intuitive, visual explanations of the model's decision mechanisms. To learn SGTs from data, we propose ShapeCART, an efficient induction algorithm for SGTs. We further extend the SGT framework to bivariate shape functions (S2GT) and multi-way trees (SGTK), and present Shape2CART and ShapeCARTK, extensions to ShapeCART for learning S2GTs and SGTKs, respectively. Experiments on various datasets show that SGTs achieve superior performance with reduced model size compared to traditional axis-aligned linear trees.

Existing approaches usually assume shared generic knowledge (e.g., prototypes, spectral features) via aggregating local structures statistically to alleviate structural heterogeneity. However, imposing overly strict assumptions about the presumed correlation between structural features and the global objective often fails in generalizing to local tasks, leading to suboptimal performance. To tackle this issue, we propose a Federated Invariant Graph Learning (FedIGL) framework based on invariant learning, which effectively disrupts spurious correlations and further mines the invariant factors across different distributions. Specifically, a server-side global model is trained to capture client-agnostic subgraph patterns shared across clients, whereas client-side models specialize in client-specific subgraph patterns. Subsequently, without compromising privacy, we propose a novel Bi-Gradient Regularization strategy that introduces gradient constraints to guide the model in identifying client-agnostic and client-specific subgraph patterns for better graph representations. Extensive experiments on graph-level clustering and classification tasks demonstrate the superiority of FedIGL against its competitors.

However, existing deep constrained clustering (DCC) methods are either limited by anchors inherent in end-to-end modeling or struggle with learning discriminative Euclidean embedding, restricting their scalability and real-world applicability. To avoid their respective pitfalls, we propose a novel angular constraint embedding approach for DCC, termed SpherePair. Using the SpherePair loss with a geometric formulation, our method faithfully encodes pairwise constraints and leads to embeddings that are clustering-friendly in angular space, effectively separating representation learning from clustering. SpherePair preserves pairwise relations without conflict, removes the need to specify the exact number of clusters, generalizes to unseen data, enables rapid inference of the number of clusters, and is supported by rigorous theoretical guarantees. Comparative evaluations with stateof-the-art DCC methods on diverse benchmarks, along with empirical validation of theoretical insights, confirm its superior performance, scalability, and overall real-world effectiveness. Code is available at our repository.

Spatial transcriptomics (ST) technologies provide gene expression measurements with spatial resolution, enabling the dissection of tissue structure and function. A fundamental challenge in ST analysis is clustering spatial spots into coherent functional regions. While existing models effectively integrate expression and spatial signals, they largely overlook sequence-level biological priors encoded in the DNA sequences of expressed genes. To bridge this gap, we propose SAINT (Sequence-Aware Integration for Nucleotide-informed Transcriptomics), a unified framework that augments spatial representation learning with nucleotide-derived features. We construct sequence-augmented datasets across 14 tissue sections from three widely used ST benchmarks (DLPFC, HBC, and MBA), retrieving reference DNA sequences for each expressed gene and encoding them using a pretrained Nucleotide Transformer. For each spot, gene-level embeddings are aggregated via expression-weighted and attention-based pooling, then fused with spatial-expression representations through a late fusion module. Extensive experiments demonstrate that SAINT consistently improves clustering performance across multiple datasets.

Due to its powerful capability of self-supervised representation learning and clustering, contrastive attributed graph clustering (CAGC) has achieved great success, which mainly depends on effective data augmentation and contrastive objective setting. However, most CAGC methods utilize edges as auxiliary information to obtain node-level embedding representation and only focus on node-level embedding augmentation. This approach overlooks edge-level embedding augmentation and the interactions between node-level and edge-level embedding augmentations across various granularity. Moreover, they often treat all contrastive sample pairs equally, neglecting the significant differences between hard and easy positivenegative sample pairs, which ultimately limits their discriminative capability. To tackle these issues, a novel robust attributed graph clustering (RAGC), incorporating hybrid-collaborative augmentation (HCA) and contrastive sample adaptivedifferential awareness (CSADA), is proposed. First, node-level and edge-level embedding representations and augmentations are simultaneously executed to establish a more comprehensive similarity measurement criterion for subsequent contrastive learning.

Current Large Language Models (LLMs) are confronted with overwhelming information volume when comprehending long-form documents. This challenge raises the imperative of a cohesive memory module, which can elevate vanilla LLMs into autonomous reading agents. Despite the emergence of some heuristic approaches, a systematic design principle remains absent. To fill this void, we draw inspiration from Jean Piaget's Constructivist Theory, illuminating three traits of the agentic memory--structured schemata, flexible assimilation, and dynamic accommodation.