With the advent of Big Data, algorithms that try to extract information from them are ubiquitous. Clustering algorithms are a subcategory of Machine Learning algorithms with a wide range of applications. Notions like closeness, distance and shape are central to clustering. It is then natural to try to use ideas and techniques from topology to improve clustering algorithms. This chapter examines some ways on how this could happen. In Section 2 a brief introduction to the clustering task is presented. In Subsection 2.1 a definition is presented along with notation.

Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat flow-based local network dynamics. In fact, we demonstrate a procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the heat flow dynamics for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. We validate our approach by successful applications to real-world data from a wide array of application domains, including computer science, genetics, climatology and economics. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.

In multivariate time series systems, it has been observed that certain groups of variables partially lead the evolution of the system, while other variables follow this evolution with a time delay; the result is a lead-lag structure amongst the time series variables. In this paper, we propose a method for the detection of lead-lag clusters of time series in multivariate systems. We demonstrate that the web of pairwise lead-lag relationships between time series can be helpfully construed as a directed network, for which there exist suitable algorithms for the detection of pairs of lead-lag clusters with high pairwise imbalance. Within our framework, we consider a number of choices for the pairwise lead-lag metric and directed network clustering components. Our framework is validated on both a synthetic generative model for multivariate lead-lag time series systems and daily real-world US equity prices data. We showcase that our method is able to detect statistically significant lead-lag clusters in the US equity market. We study the nature of these clusters in the context of the empirical finance literature on lead-lag relations and demonstrate how these can be used for the construction of predictive financial signals.

Topic taxonomies, which represent the latent topic (or category) structure of document collections, provide valuable knowledge of contents in many applications such as web search and information filtering. Recently, several unsupervised methods have been developed to automatically construct the topic taxonomy from a text corpus, but it is challenging to generate the desired taxonomy without any prior knowledge. In this paper, we study how to leverage the partial (or incomplete) information about the topic structure as guidance to find out the complete topic taxonomy. We propose a novel framework for topic taxonomy completion, named TaxoCom, which recursively expands the topic taxonomy by discovering novel sub-topic clusters of terms and documents. To effectively identify novel topics within a hierarchical topic structure, TaxoCom devises its embedding and clustering techniques to be closely-linked with each other: (i) locally discriminative embedding optimizes the text embedding space to be discriminative among known (i.e., given) sub-topics, and (ii) novelty adaptive clustering assigns terms into either one of the known sub-topics or novel sub-topics. Our comprehensive experiments on two real-world datasets demonstrate that TaxoCom not only generates the high-quality topic taxonomy in terms of term coherency and topic coverage but also outperforms all other baselines for a downstream task.

Supervised deep learning has shown state-of-the-art performance for medical image segmentation across different applications, including histopathology and cancer research; however, the manual annotation of such data is extremely laborious. In this work, we explore the use of superpixel approaches to compute a pre-segmentation of HER2 stained images for breast cancer diagnosis that facilitates faster manual annotation and correction in a second step. Four methods are compared: Standard Simple Linear Iterative Clustering (SLIC) as a baseline, a domain adapted SLIC, and superpixels based on feature embeddings of a pretrained ResNet-50 and a denoising autoencoder. To tackle oversegmentation, we propose to hierarchically merge superpixels, based on their content in the respective feature space. When evaluating the approaches on fully manually annotated images, we observe that the autoencoder-based superpixels achieve a 23% increase in boundary F1 score compared to the baseline SLIC superpixels. Furthermore, the boundary F1 score increases by 73% when hierarchical clustering is applied on the adapted SLIC and the autoencoder-based superpixels. These evaluations show encouraging first results for a pre-segmentation for efficient manual refinement without the need for an initial set of annotated training data.

We consider the problem of multiway clustering in the presence of unknown degree heterogeneity. Such data problems arise commonly in applications such as recommendation system, neuroimaging, community detection, and hypergraph partitions in social networks. The allowance of degree heterogeneity provides great flexibility in clustering models, but the extra complexity poses significant challenges in both statistics and computation. Here, we develop a degree-corrected tensor block model with estimation accuracy guarantees. We present the phase transition of clustering performance based on the notion of angle separability, and we characterize three signal-to-noise regimes corresponding to different statistical-computational behaviors. In particular, we demonstrate that an intrinsic statistical-to-computational gap emerges only for tensors of order three or greater. Further, we develop an efficient polynomial-time algorithm that provably achieves exact clustering under mild signal conditions. The efficacy of our procedure is demonstrated through two data applications, one on human brain connectome project, and another on Peru Legislation network dataset.

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We address general-shaped clustering problems under very weak parametric assumptions with a two-step hybrid robust clustering algorithm based on trimmed k-means and hierarchical agglomeration. The algorithm has low computational complexity and effectively identifies the clusters also in presence of data contamination. We also present natural generalizations of the approach as well as an adaptive procedure to estimate the amount of contamination in a data-driven fashion. Our proposal outperforms state-of-the-art robust, model-based methods in our numerical simulations and real-world applications related to color quantization for image analysis, human mobility patterns based on GPS data, biomedical images of diabetic retinopathy, and functional data across weather stations.

Variability in social app usage across regions results in a high skew of the quantity and the quality of check-in data collected, which in turn is a challenge for effective location recommender systems. In this paper, we present Axolotl (Automated cross Location-network Transfer Learning), a novel method aimed at transferring location preference models learned in a data-rich region to significantly boost the quality of recommendations in a data-scarce region. Axolotl predominantly deploys two channels for information transfer, (1) a meta-learning based procedure learned using location recommendation as well as social predictions, and (2) a lightweight unsupervised cluster-based transfer across users and locations with similar preferences. Both of these work together synergistically to achieve improved accuracy of recommendations in data-scarce regions without any prerequisite of overlapping users and with minimal fine-tuning. We build Axolotl on top of a twin graph-attention neural network model used for capturing the user- and location-conditioned influences in a user-mobility graph for each region. We conduct extensive experiments on 12 user mobility datasets across the U.S., Japan, and Germany, using 3 as source regions and 9 of them (that have much sparsely recorded mobility data) as target regions. Empirically, we show that Axolotl achieves up to 18% better recommendation performance than the existing state-of-the-art methods across all metrics.

Despite the impressive clustering performance and efficiency in characterizing both the relationship between data and cluster structure, existing graph-based multi-view clustering methods still have the following drawbacks. They suffer from the expensive time burden due to both the construction of graphs and eigen-decomposition of Laplacian matrix, and fail to explore the cluster structure of large-scale data. Moreover, they require a post-processing to get the final clustering, resulting in suboptimal performance. Furthermore, rank of the learned view-consensus graph cannot approximate the target rank. In this paper, drawing the inspiration from the bipartite graph, we propose an effective and efficient graph learning model for multi-view clustering. Specifically, our method exploits the view-similar between graphs of different views by the minimization of tensor Schatten p-norm, which well characterizes both the spatial structure and complementary information embedded in graphs of different views. We learn view-consensus graph with adaptively weighted strategy and connectivity constraint such that the connected components indicates clusters directly. Our proposed algorithm is time-economical and obtains the stable results and scales well with the data size. Extensive experimental results indicate that our method is superior to state-of-the-art methods.