We introduce Cayley transform ellipsoid fitting (CTEF), an algorithm that uses the Cayley transform to fit ellipsoids to noisy data in any dimension. Unlike many ellipsoid fitting methods, CTEF is ellipsoid specific, meaning it always returns elliptic solutions, and can fit arbitrary ellipsoids. It also significantly outperforms other fitting methods when data are not uniformly distributed over the surface of an ellipsoid. Inspired by growing calls for interpretable and reproducible methods in machine learning, we apply CTEF to dimension reduction, data visualization, and clustering in the context of cell cycle and circadian rhythm data and several classical toy examples. Since CTEF captures global curvature, it extracts nonlinear features in data that other machine learning methods fail to identify. For example, on the clustering examples CTEF outperforms 10 popular algorithms.

The study of complex networks has significantly advanced our understanding of community structures which serves as a crucial feature of real-world graphs. Detecting communities in graphs is a challenging problem with applications in sociology, biology, and computer science. Despite the efforts of an interdisciplinary community of scientists, a satisfactory solution to this problem has not yet been achieved. This review article delves into the topic of community detection in graphs, which serves as a crucial role in understanding the organization and functioning of complex systems. We begin by introducing the concept of community structure, which refers to the arrangement of vertices into clusters, with strong internal connections and weaker connections between clusters. Then, we provide a thorough exposition of various community detection methods, including a new method designed by us. Additionally, we explore real-world applications of community detection in diverse networks. In conclusion, this comprehensive review provides a deep understanding of community detection in graphs. It serves as a valuable resource for researchers and practitioners in multiple disciplines, offering insights into the challenges, methodologies, and applications of community detection in complex networks.

A popular heuristic method for improving clustering results is to apply dimensionality reduction before running clustering algorithms. It has been observed that spectral-based dimensionality reduction tools, such as PCA or SVD, improve the performance of clustering algorithms in many applications. This phenomenon indicates that spectral method not only serves as a dimensionality reduction tool, but also contributes to the clustering procedure in some sense. It is an interesting question to understand the behavior of spectral steps in clustering problems. As an initial step in this direction, this paper studies the power of vanilla-SVD algorithm in the stochastic block model (SBM). We show that, in the symmetric setting, vanilla-SVD algorithm recovers all clusters correctly. This result answers an open question posed by Van Vu (Combinatorics Probability and Computing, 2018) in the symmetric setting.

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.

In this study, we delve into the dynamics of Wordle using data analysis and machine learning. Our analysis initially focused on the correlation between the date and the number of submitted results. Due to initial popularity bias, we modeled stable data using an ARIMAX model with coefficient values of 9, 0, 2, and weekdays/weekends as the exogenous variable. We found no significant relationship between word attributes and hard mode results. To predict word difficulty, we employed a Backpropagation Neural Network, overcoming overfitting via feature engineering. We also used K-means clustering, optimized at five clusters, to categorize word difficulty numerically. Our findings indicate that on March 1st, 2023, around 12,884 results will be submitted and the word "eerie" averages 4.8 attempts, falling into the hardest difficulty cluster. We further examined the percentage of loyal players and their propensity to undertake daily challenges. Our models underwent rigorous sensitivity analyses, including ADF, ACF, PACF tests, and cross-validation, confirming their robustness. Overall, our study provides a predictive framework for Wordle gameplay based on date or a given five-letter word. Results have been summarized and submitted to the Puzzle Editor of the New York Times.

Combinatorial inverse problems in high energy physics span enormous algorithmic challenges. This work presents a new deep learning driven clustering algorithm that utilizes a space-time non-local trainable graph constructor, a graph neural network, and a set transformer. The model is trained with loss functions at the graph node, edge and object level, including contrastive learning and meta-supervision. The algorithm can be applied to problems such as charged particle tracking, calorimetry, pile-up discrimination, jet physics, and beyond. We showcase the effectiveness of this cutting-edge AI approach through particle tracking simulations. The code is available online.

Clustering clients into groups that exhibit relatively homogeneous data distributions represents one of the major means of improving the performance of federated learning (FL) in non-independent and identically distributed (non-IID) data settings. Yet, the applicability of current state-of-the-art approaches remains limited as these approaches cluster clients based on information, such as the evolution of local model parameters, that is only obtainable through actual on-client training. On the other hand, there is a need to make FL models available to clients who are not able to perform the training themselves, as they do not have the processing capabilities required for training, or simply want to use the model without participating in the training. Furthermore, the existing alternative approaches that avert the training still require that individual clients have a sufficient amount of labeled data upon which the clustering is based, essentially assuming that each client is a data annotator. In this paper, we present REPA, an approach to client clustering in non-IID FL settings that requires neither training nor labeled data collection. REPA uses a novel supervised autoencoder-based method to create embeddings that profile a client's underlying data-generating processes without exposing the data to the server and without requiring local training. Our experimental analysis over three different datasets demonstrates that REPA delivers state-of-the-art model performance while expanding the applicability of cluster-based FL to previously uncovered use cases.

The present study proposes clustering techniques for designing demand response (DR) programs for commercial and residential prosumers. The goal is to alter the consumption behavior of the prosumers within a distributed energy community in Italy. This aggregation aims to: a) minimize the reverse power flow at the primary substation, occuring when generation from solar panels in the local grid exceeds consumption, and b) shift the system wide peak demand, that typically occurs during late afternoon. Regarding the clustering stage, we consider daily prosumer load profiles and divide them across the extracted clusters. Three popular machine learning algorithms are employed, namely k-means, k-medoids and agglomerative clustering. We evaluate the methods using multiple metrics including a novel metric proposed within this study, namely peak performance score (PPS). The k-means algorithm with dynamic time warping distance considering 14 clusters exhibits the highest performance with a PPS of 0.689. Subsequently, we analyze each extracted cluster with respect to load shape, entropy, and load types. These characteristics are used to distinguish the clusters that have the potential to serve the optimization objectives by matching them to proper DR schemes including time of use, critical peak pricing, and real-time pricing. Our results confirm the effectiveness of the proposed clustering algorithm in generating meaningful flexibility clusters, while the derived DR pricing policy encourages consumption during off-peak hours. The developed methodology is robust to the low availability and quality of training datasets and can be used by aggregator companies for segmenting energy communities and developing personalized DR policies.

We propose a cluster-based method to detect and locate eavesdropping events in optical line systems characterized by small power losses. Our findings indicate that detecting such subtle losses from eavesdropping can be accomplished solely through optical performance monitoring (OPM) data collected at the receiver. On the other hand, the localization of such events can be effectively achieved by leveraging in-line OPM data.

Thanks to their dependency structure, non-parametric Hidden Markov Models (HMMs) are able to handle model-based clustering without specifying group distributions. The aim of this work is to study the Bayes risk of clustering when using HMMs and to propose associated clustering procedures. We first give a result linking the Bayes risk of classification and the Bayes risk of clustering, which we use to identify the key quantity determining the difficulty of the clustering task. We also give a proof of this result in the i.i.d. framework, which might be of independent interest. Then we study the excess risk of the plugin classifier. All these results are shown to remain valid in the online setting where observations are clustered sequentially. Simulations illustrate our findings.