In this article, we will discuss hierarchical clustering algorithms in unsupervised machine learning. This algorithm is based on the splitting and merging of nested clusters. The linkage criteria to merge the clusters based on distance metric as shown below with a bottom-up approach. The linkage criteria give different clusters at different time speeds. Single linkage is not good in noisy data and ward linkage does not give proper cluster because the distance is not varied thus but good in properly balanced clusters and if we do not consider euclidean then average linkage can be used for clustering. The next parameter comes is connectivity that connects or merges the clusters based on the connectivity matrix.

Deep clustering has the potential to learn a strong representation and hence better clustering performance compared to traditional clustering methods such as $k$-means and spectral clustering. However, this strong representation learning ability may make the clustering unfair by discovering surrogates for protected information which we empirically show in our experiments. In this work, we study a general notion of group-level fairness for both binary and multi-state protected status variables (PSVs). We begin by formulating the group-level fairness problem as an integer linear programming formulation whose totally unimodular constraint matrix means it can be efficiently solved via linear programming. We then show how to inject this solver into a discriminative deep clustering backbone and hence propose a refinement learning algorithm to combine the clustering goal with the fairness objective to learn fair clusters adaptively. Experimental results on real-world datasets demonstrate that our model consistently outperforms state-of-the-art fair clustering algorithms. Our framework shows promising results for novel clustering tasks including flexible fairness constraints, multi-state PSVs and predictive clustering.

Analysis of large observational data sets generated by a reactive system is a common challenge in debugging system failures and determining their root cause. One of the major problems is that these observational data suffer from survivorship bias. Examples include analyzing traffic logs from networks, and simulation logs from circuit design. In such applications, users want to detect non-spurious correlations from observational data and obtain actionable insights about them. In this paper, we introduce log to Neuro-symbolic (Log2NS), a framework that combines probabilistic analysis from machine learning (ML) techniques on observational data with certainties derived from symbolic reasoning on an underlying formal model. We apply the proposed framework to network traffic debugging by employing the following steps. To detect patterns in network logs, we first generate global embedding vector representations of entities such as IP addresses, ports, and applications. Next, we represent large log flow entries as clusters that make it easier for the user to visualize and detect interesting scenarios that will be further analyzed. To generalize these patterns, Log2NS provides an ability to query from static logs and correlation engines for positive instances, as well as formal reasoning for negative and unseen instances. By combining the strengths of deep learning and symbolic methods, Log2NS provides a very powerful reasoning and debugging tool for log-based data. Empirical evaluations on a real internal data set demonstrate the capabilities of Log2NS.

In the application of data clustering to human-centric decision-making systems, such as loan applications and advertisement recommendations, the clustering outcome might discriminate against people across different demographic groups, leading to unfairness. A natural conflict occurs between the cost of clustering (in terms of distance to cluster centers) and the balance representation of all demographic groups across the clusters, leading to a bi-objective optimization problem that is nonconvex and nonsmooth. To determine the complete trade-off between these two competing goals, we design a novel stochastic alternating balance fair $k$-means (SAfairKM) algorithm, which consists of alternating classical mini-batch $k$-means updates and group swap updates. The number of $k$-means updates and the number of swap updates essentially parameterize the weight put on optimizing each objective function. Our numerical experiments show that the proposed SAfairKM algorithm is robust and computationally efficient in constructing well-spread and high-quality Pareto fronts both on synthetic and real datasets. Moreover, we propose a novel companion algorithm, the stochastic alternating bi-objective gradient descent (SA2GD) algorithm, which can handle a smooth version of the considered bi-objective fair $k$-means problem, more amenable for analysis. A sublinear convergence rate of $\mathcal{O}(1/T)$ is established under strong convexity for the determination of a stationary point of a weighted sum of the two functions parameterized by the number of steps or updates on each function.

Dense networks with weighted connections often exhibit a community like structure, where although most nodes are connected to each other, different patterns of edge weights may emerge depending on each node's community membership. We propose a new framework for generating and estimating dense weighted networks with potentially different connectivity patterns across different communities. The proposed model relies on a particular class of functions which map individual node characteristics to the edges connecting those nodes, allowing for flexibility while requiring a small number of parameters relative to the number of edges. By leveraging the estimation techniques, we also develop a bootstrap methodology for generating new networks on the same set of vertices, which may be useful in circumstances where multiple data sets cannot be collected. Performance of these methods are analyzed in theory, simulations, and real data.

Data summarizations are a valuable tool to derive knowledge from large data streams and have proven their usefulness in a great number of applications. Summaries can be found by optimizing submodular functions. These functions map subsets of data to real values, which indicate their "representativeness" and which should be maximized to find a diverse summary of the underlying data. In this paper, we studied Exemplar-based clustering as a submodular function and provide a GPU algorithm to cope with its high computational complexity. We show, that our GPU implementation provides speedups of up to 72x using single-precision and up to 452x using half-precision computation compared to conventional CPU algorithms. We also show, that the GPU algorithm not only provides remarkable runtime benefits with workstation-grade GPUs but also with low-power embedded computation units for which speedups of up to 35x are possible. Furthermore, we apply our algorithm to real-world data from injection molding manufacturing processes and discuss how found summaries help with steering this specific process to cut costs and reduce the manufacturing of bad parts. Beyond pure speedup considerations, we show, that our approach can provide summaries within reasonable time frames for this kind of industrial, real-world data.

Recent work on explainable clustering allows describing clusters when the features are interpretable. However, much modern machine learning focuses on complex data such as images, text, and graphs where deep learning is used but the raw features of data are not interpretable. This paper explores a novel setting for performing clustering on complex data while simultaneously generating explanations using interpretable tags. We propose deep descriptive clustering that performs sub-symbolic representation learning on complex data while generating explanations based on symbolic data. We form good clusters by maximizing the mutual information between empirical distribution on the inputs and the induced clustering labels for clustering objectives. We generate explanations by solving an integer linear programming that generates concise and orthogonal descriptions for each cluster. Finally, we allow the explanation to inform better clustering by proposing a novel pairwise loss with self-generated constraints to maximize the clustering and explanation module's consistency. Experimental results on public data demonstrate that our model outperforms competitive baselines in clustering performance while offering high-quality cluster-level explanations.

Deep learning on graphs has recently achieved remarkable success on a variety of tasks while such success relies heavily on the massive and carefully labeled data. However, precise annotations are generally very expensive and time-consuming. To address this problem, self-supervised learning (SSL) is emerging as a new paradigm for extracting informative knowledge through well-designed pretext tasks without relying on manual labels. In this survey, we extend the concept of SSL, which first emerged in the fields of computer vision and natural language processing, to present a timely and comprehensive review of the existing SSL techniques for graph data. Specifically, we divide existing graph SSL methods into three categories: contrastive, generative, and predictive. More importantly, unlike many other surveys that only provide a high-level description of published research, we present an additional mathematical summary of the existing works in a unified framework. Furthermore, to facilitate methodological development and empirical comparisons, we also summarize the commonly used datasets, evaluation metrics, downstream tasks, and open-source implementations of various algorithms. Finally, we discuss the technical challenges and potential future directions for improving graph self-supervised learning.

In this post, I introduce a new Python package to generate clustergrams from clustering solutions. The library has been developed as part of the Urban Grammar research project, and it is compatible with scikit-learn and GPU-enabled libraries such as cuML or cuDF within RAPIDS.AI. When we want to do some cluster analysis to identify groups in our data, we often use algorithms like K-Means, which require the specification of a number of clusters. But the issue is that we usually don't know how many clusters there are. There are many methods on how to determine the correct number, like silhouettes or elbow plot, to name a few.

Convex clustering is an attractive clustering algorithm with favorable properties such as efficiency and optimality owing to its convex formulation. It is thought to generalize both k-means clustering and agglomerative clustering. However, it is not known whether convex clustering preserves desirable properties of these algorithms. A common expectation is that convex clustering may learn difficult cluster types such as non-convex ones. Current understanding of convex clustering is limited to only consistency results on well-separated clusters. We show new understanding of its solutions. We prove that convex clustering can only learn convex clusters. We then show that the clusters have disjoint bounding balls with significant gaps. We further characterize the solutions, regularization hyperparameters, inclusterable cases and consistency.