Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.

Agglomerative hierarchical clustering is one of the most widely used approaches for exploring how observations in a dataset relate to each other. However, its greedy nature makes it highly sensitive to small perturbations in the data, often producing different clustering results and making it difficult to separate genuine structure from spurious patterns. In this paper, we show how randomizing hierarchical clustering can be useful not just for measuring stability but also for designing valid hypothesis testing procedures based on the clustering results. We propose a simple randomization scheme together with a method for constructing a valid p-value at each node of the hierarchical clustering dendrogram that quantifies evidence against performing the greedy merge. Our test controls the Type I error rate, works with any hierarchical linkage without case-specific derivations, and simulations show it is substantially more powerful than existing selective inference approaches. To demonstrate the practical utility of our p-values, we develop an adaptive $α$-spending procedure that estimates the number of clusters, with a probabilistic guarantee on overestimation. Experiments on simulated and real data show that this estimate yields powerful clustering and can be used, for example, to assess clustering stability across multiple runs of the randomized algorithm.

Hierarchical clustering is a fundamental machine-learning technique for grouping data points into dendrograms. However, existing hierarchical clustering methods encounter two primary challenges: 1) Most methods specify dendrograms without a global objective. 2) Graph-based methods often neglect the significance of graph structure, optimizing objectives on complete or static predefined graphs. In this work, we propose Hyperbolic Continuous Structural Entropy neural networks, namely HypCSE, for structure-enhanced continuous hierarchical clustering. Our key idea is to map data points in the hyperbolic space and minimize the relaxed continuous structural entropy (SE) on structure-enhanced graphs. Specifically, we encode graph vertices in hyperbolic space using hyperbolic graph neural networks and minimize approximate SE defined on graph embeddings. To make the SE objective differentiable for optimization, we reformulate it into a function using the lowest common ancestor (LCA) on trees and then relax it into continuous SE (CSE) by the analogy of hyperbolic graph embeddings and partitioning trees. To ensure a graph structure that effectively captures the hierarchy of data points for CSE calculation, we employ a graph structure learning (GSL) strategy that updates the graph structure during training. Extensive experiments on seven datasets demonstrate the superior performance of HypCSE.

Neural Information Processing Systems http://nips.cc/

We thank the reviewers for the time they spent evaluating our manuscript and for their valuable comments. We agree that having theoretical guarantees would be a big plus. As for scalability, the bottleneck of our method is the single-linkage algorithm. Similarly to Monath et al. (NeurIPS 2017), our idea consists Given the significant body of additional material, we feel that this topic is best left to a future publication. Line 8,56,70,93: I would suggest a more cautious usage of the word "equivalent".

We develop a unified statistical framework for softmax-gated Gaussian mixture of experts (SGMoE) that addresses three long-standing obstacles in parameter estimation and model selection: (i) non-identifiability of gating parameters up to common translations, (ii) intrinsic gate-expert interactions that induce coupled differential relations in the likelihood, and (iii) the tight numerator-denominator coupling in the softmax-induced conditional density. Our approach introduces Voronoi-type loss functions aligned with the gate-partition geometry and establishes finite-sample convergence rates for the maximum likelihood estimator (MLE). In over-specified models, we reveal a link between the MLE's convergence rate and the solvability of an associated system of polynomial equations characterizing near-nonidentifiable directions. For model selection, we adapt dendrograms of mixing measures to SGMoE, yielding a consistent, sweep-free selector of the number of experts that attains pointwise-optimal parameter rates under overfitting while avoiding multi-size training. Simulations on synthetic data corroborate the theory, accurately recovering the expert count and achieving the predicted rates for parameter estimation while closely approximating the regression function. Under model misspecification (e.g., $ε$-contamination), the dendrogram selection criterion is robust, recovering the true number of mixture components, while the Akaike information criterion, the Bayesian information criterion, and the integrated completed likelihood tend to overselect as sample size grows. On a maize proteomics dataset of drought-responsive traits, our dendrogram-guided SGMoE selects two experts, exposes a clear mixing-measure hierarchy, stabilizes the likelihood early, and yields interpretable genotype-phenotype maps, outperforming standard criteria without multi-size training.

In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure.

Neural Information Processing Systems http://nips.cc/

We thank the reviewers for the time they spent evaluating our manuscript and for their valuable comments. We agree that having theoretical guarantees would be a big plus. As for scalability, the bottleneck of our method is the single-linkage algorithm. Similarly to Monath et al. (NeurIPS 2017), our idea consists Given the significant body of additional material, we feel that this topic is best left to a future publication. Line 8,56,70,93: I would suggest a more cautious usage of the word "equivalent".