Persistence diagrams, a key tool in the field of Topological Data Analysis, concisely represent the topology of a point cloud. Most current methods to integrate persistence diagrams into machine learning either require prior knowledge of a ground-truth topology or map them into a feature vector, offering a trade-off between information loss and invoking the curse of dimensionality. In this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams, enabling unsupervised learning that automatically captures the topological structure of data, with no prior knowledge or additional processing of persistence diagrams. We prove the same convergence guarantees as traditional FCM clustering: that any convergent subsequence of our algorithm tends to a local minimum or saddle point of the cost function. We end by presenting experiments that demonstrate our algorithm can successfully cluster transformed datasets from materials science where comparable Wasserstein barycentre clustering algorithms fail, whilst also running at least an order of magnitude faster.

We present a visualization algorithm based on a novel unsupervised Siamese neural network training regime and loss function, called Differentiating Embedding Networks (DEN). The Siamese neural network finds differentiating or similar features between specific pairs of samples in a dataset, and uses these features to embed the dataset in a lower dimensional space where it can be visualized. Unlike existing visualization algorithms such as UMAP or $t$-SNE, DEN is parametric, meaning it can be interpreted by techniques such as SHAP. To interpret DEN, we create an end-to-end parametric clustering algorithm on top of the visualization, and then leverage SHAP scores to determine which features in the sample space are important for understanding the structures shown in the visualization based on the clusters found. We compare DEN visualizations with existing techniques on a variety of datasets, including image and scRNA-seq data. We then show that our clustering algorithm performs similarly to the state of the art despite not having prior knowledge of the number of clusters, and sets a new state of the art on FashionMNIST. Finally, we demonstrate finding differentiating features of a dataset. Code available at https://github.com/isaacrob/DEN

In real-world classification problems, pairwise supervision (i.e., a pair of patterns with a binary label indicating whether they belong to the same class or not) can often be obtained at a lower cost than ordinary class labels. Similarity learning is a general framework to utilize such pairwise supervision to elicit useful representations by inferring the relationship between two data points, which encompasses various important preprocessing tasks such as metric learning, kernel learning, graph embedding, and contrastive representation learning. Although elicited representations are expected to perform well in downstream tasks such as classification, little theoretical insight has been given in the literature so far. In this paper, we reveal that a specific formulation of similarity learning is strongly related to the objective of binary classification, which spurs us to learn a binary classifier without ordinary class labels---by fitting the product of real-valued prediction functions of pairwise patterns to their similarity. Our formulation of similarity learning does not only generalize many existing ones, but also admits an excess risk bound showing an explicit connection to classification. Finally, we empirically demonstrate the practical usefulness of the proposed method on benchmark datasets.

Topic detection is a challenging task, especially without knowing the exact number of topics. In this paper, we present a novel approach based on neural network to detect topics in the micro-blogging dataset. We use an unsupervised neural sentence embedding model to map the blogs to an embedding space. Our model is a weighted power mean word embedding model, and the weights are calculated by attention mechanism. Experimental result shows our embedding method performs better than baselines in sentence clustering. In addition, we propose an improved clustering algorithm referred as relationship-aware DBSCAN (RADBSCAN). It can discover topics from a micro-blogging dataset, and the topic number depends on dataset character itself. Moreover, in order to solve the problem of parameters sensitive, we take blog forwarding relationship as a bridge of two independent clusters. Finally, we validate our approach on a dataset from sina micro-blog. The result shows that we can detect all the topics successfully and extract keywords in each topic.

The implementation of efficient demand response (DR) programs for household electricity consumption would benefit from data-driven methods capable of simulating the impact of different tariffs schemes. This paper proposes a novel method based on conditional variational autoencoders (CVAE) to generate, from an electricity tariff profile combined with exogenous weather and calendar variables, daily consumption profiles of consumers segmented in different clusters. First, a large set of consumers is gathered into clusters according to their consumption behavior and price-responsiveness. The clustering method is based on a causality model that measures the effect of a specific tariff on the consumption level. Then, daily electrical energy consumption profiles are generated for each cluster with CVAE. This non-parametric approach is compared to a semi-parametric data generator based on generalized additive models and that uses prior knowledge of energy consumption. Experiments in a publicly available data set show that, the proposed method presents comparable performance to the semi-parametric one when it comes to generating the average value of the original data. The main contribution from this new method is the capacity to reproduce rebound and side effects in the generated consumption profiles. Indeed, the application of a special electricity tariff over a time window may also affect consumption outside this time window. Another contribution is that the clustering approach segments consumers according to their daily consumption profile and elasticity to tariff changes. These two results combined are very relevant for an ex-ante testing of future DR policies by system operators, retailers and energy regulators.

Loss-based clustering methods, such as k-means and its variants, are standard tools for finding groups in data. However, the lack of quantification of uncertainty in the estimated clusters is a disadvantage. Model-based clustering based on mixture models provides an alternative, but such methods face computational problems and large sensitivity to the choice of kernel. This article proposes a generalized Bayes framework that bridges between these two paradigms through the use of Gibbs posteriors. In conducting Bayesian updating, the log likelihood is replaced by a loss function for clustering, leading to a rich family of clustering methods. The Gibbs posterior represents a coherent updating of Bayesian beliefs without needing to specify a likelihood for the data, and can be used for characterizing uncertainty in clustering. We consider losses based on Bregman divergence and pairwise similarities, and develop efficient deterministic algorithms for point estimation along with sampling algorithms for uncertainty quantification. Several existing clustering algorithms, including k-means, can be interpreted as generalized Bayes estimators under our framework, and hence we provide a method of uncertainty quantification for these approaches.

In order to help physicists to expand their knowledge of the climate in the Lesser Antilles, we aim to identify the spatio-temporal configurations using clustering analysis on wind speed and cumulative rainfall datasets. But we show that using the L2 norm in conventional clustering methods as K-Means (KMS) and Hierarchical Agglomerative Clustering (HAC) can induce undesirable effects. So, we propose to replace Euclidean distance (L2) by a dissimilarity measure named Expert Deviation (ED). Based on the symmetrized Kullback-Leibler divergence, the ED integrates the properties of the observed physical parameters and climate knowledge. This measure helps comparing histograms of four patches, corresponding to geographical zones, that are influenced by atmospheric structures. The combined evaluation of the internal homogeneity and the separation of the clusters obtained using ED and L2 was performed. The results, which are compared using the silhouette index, show five clusters with high indexes. For the two available datasets one can see that, unlike KMS-L2, KMS-ED discriminates the daily situations favorably, giving more physical meaning to the clusters discovered by the algorithm. The effect of patches is observed in the spatial analysis of representative elements for KMS-ED. The ED is able to produce different configurations which makes the usual atmospheric structures clearly identifiable. Atmospheric physicists can interpret the locations of the impact of each cluster on a specific zone according to atmospheric structures. KMS-L2 does not lead to such an interpretability, because the situations represented are spatially quite smooth. This climatological study illustrates the advantage of using ED as a new approach.

A common distinction in fair machine learning, in particular in fair classification, is between group fairness and individual fairness. In the context of clustering, group fairness has been studied extensively in recent years; however, individual fairness for clustering has hardly been explored. In this paper, we propose a natural notion of individual fairness for clustering. Our notion asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. We study several questions related to our proposed notion of individual fairness. On the negative side, we show that deciding whether a given data set allows for such an individually fair clustering in general is NP-hard. On the positive side, for the special case of a data set lying on the real line, we propose an efficient dynamic programming approach to find an individually fair clustering. For general data sets, we investigate heuristics aimed at minimizing the number of individual fairness violations and compare them to standard clustering approaches on real data sets.

We study the problem of applying spectral clustering to cluster multi-scale data, which is data whose clusters are of various sizes and densities. Traditional spectral clustering techniques discover clusters by processing a similarity matrix that reflects the proximity of objects. For multi-scale data, distance-based similarity is not effective because objects of a sparse cluster could be far apart while those of a dense cluster have to be sufficiently close. Following [16], we solve the problem of spectral clustering on multi-scale data by integrating the concept of objects' "reachability similarity" with a given distance-based similarity to derive an objects' coefficient matrix. We propose the algorithm CAST that applies trace Lasso to regularize the coefficient matrix. We prove that the resulting coefficient matrix has the "grouping effect" and that it exhibits "sparsity". We show that these two characteristics imply very effective spectral clustering. We evaluate CAST and 10 other clustering methods on a wide range of datasets w.r.t. various measures. Experimental results show that CAST provides excellent performance and is highly robust across test cases of multi-scale data.

We propose a new assumption in outlier detection: Normal data instances are commonly located in the area that there is hardly any fluctuation on data density, while outliers are often appeared in the area that there is violent fluctuation on data density. And based on this hypothesis, we apply a novel density-based approach to unsupervised outlier detection. This approach, called Quantum Clustering (QC), deals with unlabeled data processing and constructs a potential function to find the centroids of clusters and the outliers. The experiments show that the potential function could clearly find the hidden outliers in data points effectively. Besides, by using QC, we could find more subtle outliers by adjusting the parameter $\sigma$. Moreover, our approach is also evaluated on two datasets (Air Quality Detection and Darwin Correspondence Project) from two different research areas, and the results show the wide applicability of our method.