Deep clustering partitions complex high-dimensional data using deep neural networks for clustering. It involves projecting data into lower-dimensional embeddings before partitioning, which embarks unique evaluation challenges. Traditional clustering validation measures, designed for low-dimensional spaces, are problematic for deep clustering for two reasons: 1) the curse of dimensionality when applied to the high-dimensional input data, and 2) unreliable comparison of clustering results when applied to embedded data from different embedding spaces, owing to variations in training procedures and model parameter settings. This paper addresses these unresolved and often overlooked challenges in evaluating clustering within deep learning. We propose a systematic evaluation framework for internal clustering validation measures that: (1) theoretically establishes why traditional measures are ineffective when applied to input data or across disparate embedding spaces paired with partitioning outcomes; (2) identifies embedding spaces that endorse reliable evaluations by detecting groups with high agreement in ranking partitioning outcomes; and (3) develops a stable and robust scoring scheme by weighting index values computed across these identified embedding spaces. Experiments show that this new framework aligns better with external measures, effectively reducing the misguidance from the improper use of internal validation measures in deep clustering evaluation.
Istituto Italiano di Tecnologia（意大利技术研究院）；University of Trento（特伦托大学）；University of Cambridge（剑桥大学）
Determining the best partition for a dataset can be a challenging task because of 1) the lack of a priori information within an unsupervised learning framework; and 2) the absence of a unique clustering validation approach to evaluate clustering solutions. Here we present reval: a Python package that leverages stability-based relative clustering validation methods to determine best clustering solutions as the ones that best generalize to unseen data. Statistical software, both in R and Python, usually rely on internal validation metrics, such as silhouette, to select the number of clusters tha…
SLAC National Accelerator Laboratory（SLAC国家加速器实验室）
One challenge impeding the analysis of terabyte scale x-ray scattering data from the Linac Coherent Light Source LCLS, is determining the number of clusters required for the execution of traditional clustering algorithms. Here we demonstrate that previous work using bi-cross validation (BCV) to determine the number of singular vectors directly maps to the spectral clustering problem of estimating both the number of clusters and hyper parameter values. These results indicate that the process of estimating the number of clusters should not be divorced from the process of estimating other hyper p…
University of Minnesota（明尼苏达大学）；University at Buffalo, State University of New York（纽约州立大学布法罗分校）
The nowadays massive amounts of generated and communicated data present major challenges in their processing. While capable of successfully classifying nonlinearly separable objects in various settings, subspace clustering (SC) methods incur prohibitively high computational complexity when processing large-scale data. Inspired by the random sampling and consensus (RANSAC) approach to robust regression, the present paper introduces a randomized scheme for SC, termed sketching and validation (SkeVa-)SC, tailored for large-scale data. At the heart of SkeVa-SC lies a randomized scheme for approxim…
Laboratoire Informatique de Paris-Nord（巴黎北部信息学实验室）；Université Sorbonne Paris Nord（索邦巴黎北大学）；Groupe Renault SAS（雷诺集团）
Autonomous driving systems validation remains one of the biggest challenges car manufacturers must tackle in order to provide safe driverless cars. The high complexity stems from several factors: the multiplicity of vehicles, embedded systems, use cases, and the very high required level of reliability for the driving system to be at least as safe as a human driver. In order to circumvent these issues, large scale simulations reproducing this huge variety of physical conditions are intensively used to test driverless cars. Therefore, the validation step produces a massive amount of data, includ…
Selecting the Number of Clusters $K$ with a Stability Trade-off: an Internal Validation Criterion
Université Paris 1 Panthéon Sorbonne（巴黎第一大学（先贤祠-索邦大学））；Safran Aircraft Engines（赛峰飞机发动机公司）；Université Sorbonne Paris Nord（巴黎索邦北大学）；Université de Versailles/Paris-Saclay（凡尔赛大学/巴黎萨克雷大学）
Swiss Federal Institute of Technology, ETH Zurich（苏黎世联邦理工学院）
Model selection in clustering requires (i) to specify a suitable clustering principle and (ii) to control the model order complexity by choosing an appropriate number of clusters depending on the noise level in the data. We advocate an information theoretic perspective where the uncertainty in the measurements quantizes the set of data partitionings and, thereby, induces uncertainty in the solution space of clusterings. A clustering model, which can tolerate a higher level of fluctuations in the measurements than alternative models, is considered to be superior provided that the clustering sol…
Improving clustering quality evaluation in noisy Gaussian mixtures
School of Computer Science and Electronic Engineering, University of Essex（埃塞克斯大学计算机科学与电子工程学院）；Département d’informatique, Université du Québec à Montréal（魁北克大学蒙特利尔分校计算机科学系）；Mila - Quebec AI Institute（魁北克AI研究院）
In response to the need for learning tools tuned to big data analytics, the present paper introduces a framework for efficient clustering of huge sets of (possibly high-dimensional) data. Building on random sampling and consensus (RANSAC) ideas pursued earlier in a different (computer vision) context for robust regression, a suite of novel dimensionality and set-reduction algorithms is developed. The advocated sketch-and-validate (SkeVa) family includes two algorithms that rely on K-means clustering per iteration on reduced number of dimensions and/or feature vectors: The first operates in a b…
In this chapter we review the main literature related to kernel spectral clustering (KSC), an approach to clustering cast within a kernel-based optimization setting. KSC represents a least-squares support vector machine based formulation of spectral clustering described by a weighted kernel PCA objective. Just as in the classifier case, the binary clustering model is expressed by a hyperplane in a high dimensional space induced by a kernel. In addition, the multi-way clustering can be obtained by combining a set of binary decision functions via an Error Correcting Output Codes (ECOC) encoding …
Delft University of Technology（代尔夫特理工大学）；University of Cape Town（开普敦大学）；Centre for Artificial Intelligence Research（人工智能研究中心）
Clustering is frequently used in the energy domain to identify dominant electricity consumption patterns of households, which can be used to construct customer archetypes for long term energy planning. Selecting a useful set of clusters however requires extensive experimentation and domain knowledge. While internal clustering validation measures are well established in the electricity domain, they are limited for selecting useful clusters. Based on an application case study in South Africa, we present an approach for formalising implicit expert knowledge as external evaluation measures to crea…
CSTS: A Benchmark for the Discovery of Correlation Structures in Time Series Clustering
School of Computer Science, University of Bristol（计算机科学学院，布里斯托大学）；School of Engineering Mathematics and Technology, University of Bristol（工程数学与技术学院，布里斯托大学）；School of Artificial Intelligence, University of Nanjing（人工智能学院，南京大学）；College of Engineering and Architecture, University College Dublin（工程与建筑学院，都柏林大学学院）
The cost of computing the spectrum of Laplacian matrices hinders the application of spectral clustering to large data sets. While approximations recover computational tractability, they can potentially affect clustering performance. This paper proposes a practical approach to learn spectral clustering based on adaptive stochastic gradient optimization. Crucially, the proposed approach recovers the exact spectrum of Laplacian matrices in the limit of the iterations, and the cost of each iteration is linear in the number of samples. Extensive experimental validation on data sets with up to half …
Cluster Analysis with Resampling for Validation and Exploration (CARVE)
Department of Statistics, Columbia University, New York, NY, USA（哥伦比亚大学统计学系）；Center for Theoretical Neuroscience, Zuckerman Mind Brain Behavior Institute, Columbia University, New York, NY, USA（哥伦比亚大学理论神经科学中心、Zuckerman思维-大脑-行为研究所）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA（诺丁汉大学应用与计算数学与统计学系）；School of Data and Information Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA（北卡罗来纳大学夏洛特分校数据与信息科学学院）；Irving Institute for Cancer Dynamics, Columbia University, New York, NY, USA（哥伦比亚大学癌症动力学伊万·里弗斯研究所）
Singapore-MIT Alliance（新加坡-麻省理工学院联盟）；Nanyang Technological University（南洋理工大学）；National University of Singapore（新加坡国立大学）
In this paper, a similarity-driven cluster merging method is proposed for unsuper-vised fuzzy clustering. The cluster merging method is used to resolve the problem of cluster validation. Starting with an overspecified number of clusters in the data, pairs of similar clusters are merged based on the proposed similarity-driven cluster merging criterion. The similarity between clusters is calculated by a fuzzy cluster similarity matrix, while an adaptive threshold is used for merging. In addition, a modified generalized ob- jective function is used for prototype-based fuzzy clustering. The functi…
Datasets in high-dimension do not typically form clusters in their original space; the issue is worse when the number of points in the dataset is small. We propose a low-computation method to find statistically significant clustering structures in a small dataset. The method proceeds by projecting the data on a random line and seeking binary clusterings in the resulting one-dimensional data. Non-linear separations are obtained by extending the feature space using monomials of higher degrees in the original features. The statistical validity of the clustering structures obtained is tested in th…
We compare and contrast two approaches to validating a trained classifier while using all in-sample data for training. One is simultaneous validation over an organized set of hypotheses (SVOOSH), the well-known method that began with VC theory. The other is withhold and gap (WAG). WAG withholds a validation set, trains a holdout classifier on the remaining data, uses the validation data to validate that classifier, then adds the rate of disagreement between the holdout classifier and one trained using all in-sample data, which is an upper bound on the difference in error rates. We show that co…
University of Texas at Austin（得克萨斯大学奥斯汀分校）；University of Sydney（悉尼大学）
Selecting hyperparameters for unsupervised learning problems is challenging in general due to the lack of ground truth for validation. Despite the prevalence of this issue in statistics and machine learning, especially in clustering problems, there are not many methods for tuning these hyperparameters with theoretical guarantees. In this paper, we provide a framework with provable guarantees for selecting hyperparameters in a number of distinct models. We consider both the subgaussian mixture model and network models to serve as examples of i.i.d. and non-i.i.d. data. We demonstrate that the s…
We explore the performance of several automatic bandwidth selectors, originally designed for density gradient estimation, as data-based procedures for nonparametric, modal clustering. The key tool to obtain a clustering from density gradient estimators is the mean shift algorithm, which allows to obtain a partition not only of the data sample, but also of the whole space. The results of our simulation study suggest that most of the methods considered here, like cross validation and plug in bandwidth selectors, are useful for cluster analysis via the mean shift algorithm.
The selection of a validation basis from a full dataset is often required in industrial use of supervised machine learning algorithm. This validation basis will serve to realize an independent evaluation of the machine learning model. To select this basis, we propose to adopt a "design of experiments" point of view, by using statistical criteria. We show that the "support points" concept, based on Maximum Mean Discrepancy criteria, is particularly relevant. An industrial test case from the company EDF illustrates the practical interest of the methodology.
University of Tübingen（图宾根大学）；Liebherr-Elektronik GmbH（利勃海尔电子有限公司）；Scheidt MessTechnik（沙伊特测量技术公司）
We present a reformulation of the regression and classification, which aims to validate the result of a machine learning algorithm. Our reformulation simplifies the original problem and validates the result of the machine learning algorithm using the training data. Since the validation of machine learning algorithms must always be explainable, we perform our experiments with the kNN algorithm as well as with an algorithm based on conditional probabilities, which is proposed in this work. For the evaluation of our approach, three publicly available data sets were used and three classification a…
In the travel industry, online customers book their travel itinerary according to several features, like cost and duration of the travel or the quality of amenities. To provide personalized recommendations for travel searches, an appropriate segmentation of customers is required. Clustering ensemble approaches were developed to overcome well-known problems of classical clustering approaches, that each rely on a different theoretical model and can thus identify in the data space only clusters corresponding to this model. Clustering ensemble approaches combine multiple clustering results, each f…
