Bracketing Uncertainty in Clustering Under the Manifold Hypothesis
Authors: Savik Kinger, Luciano Dyballa, Steven W. Zucker
Organizations: Department of Computer Science Yale University · School of Science & Technology IE University · Depts. of Computer Science and Biomedical Engineering Wu Tsai Institute Yale University
The manifold hypothesis suggests a natural criterion for clustering: partition data according to the manifold component from which each point is drawn. Whether two components are separable depends on a geometric tradeoff: the ambient separation between components versus the largest gap in sampling. In practice, this tradeoff is rarely assessed explicitly, leading standard methods to over-commit to a single clustering assignment even when the data do not support a unique answer. We formalize this tradeoff by combining intrinsic manifold geometry (volume growth and reach) with sample-level quantities (fill distance and density), yielding a threshold phenomenon for mutual-k-nearest-neighbor graphs: when the offset-to-fill ratio exceeds a conservative upper threshold, component separation is preserved; below a lower threshold, components fuse. The gap between these thresholds defines a geometric uncertainty zone in which the number of clusters is not identifiable from the data. Nevertheless, conventional approaches still seek one: sweeping parameters (an engineering approach) or fitting a generative mixture model (a model-based approach). Rather than forcing a single estimate of the number of clusters, we propose Manifold-Based Clustering (MBC), which returns an explicit bracket interval to quantify the underlying data uncertainty. This bracket acts as an empirically calibrated diagnostic: it narrows when a single resolution is supported, widens when multiple resolutions coexist, and collapses to one when no separated structure is detectable. Empirically, we find that many real datasets lie within the uncertainty zone rather than admitting one clear answer. Our results suggest that ambiguity in cluster number is often intrinsic, and should be quantified rather than resolved.
Generative approaches to clustering provide information on geometric properties of clusters, whereas discriminative approaches provide boundaries between clusters. Ideas from both approaches are incorporated to present a fully unsupervised, probabilistic, and discriminative clustering method via a regularized mutual information objective function, wherein a mixture of mixtures of Gaussian and uniform distributions is used for formulation of the conditional model. Automatic selection of the number of components is established with the introduction of the regularizing term and a merge step, similar to those applied in reversible jump Markov chain Monte Carlo methods used in Bayesian clustering. Consequently, the turtle shell method -- a fully unsupervised clustering method capable of estimating non-linear boundary lines, automatically selecting the number of components, and capturing intuitive clusters in the presence of data abnormalities such as noise and/or irregular cluster shapes -- is introduced. We test this method on various simulated and real datasets commonly explored in clustering research, and extend the analysis to datasets arising from flow cytometry experiments.
Mackenzie R. Neal, Paul D. McNicholas, Arthur White
In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.
Despite being ubiquitous in science, clustering lacks a unified framework for quantitatively evaluating the replicability of its results. We present evaluating replicability via iterative clustering assignments (ERICA), a method for determining whether clusters can be identified reproducibly in a dataset. The pipeline computes a statistic that determines whether reproducible cluster structure is present in a dataset. Quantitative visualization methods are also introduced to characterize similarities between clusters and identify observations that may represent outliers or unstable assignments. Experiments on synthetic datasets demonstrate that ERICA successfully identifies reproducible cluster structure. In contrast, application of ERICA to three breast cancer gene-expression datasets reveals instances in which clustering solutions are not reproducible. The study underscores the importance of rigorously evaluating clustering solutions and provides a practical framework for doing so.
Siamak K. Sorooshyari, Manuel A. Rivas, Robert Tibshirani