We introduce a new prior for use in Nonparametric Bayesian Hierarchical Clustering. The prior is constructed by marginalizing out the time information of Kingman's coalescent, providing a prior over tree structures which we call the Time-Marginalized Coalescent (TMC). This allows for models which factorize the tree structure and times, providing two benefits: more flexible priors may be constructed and more efficient Gibbs type inference can be used. We demonstrate this on an example model for density estimation and show the TMC achieves competitive experimental results.

Learning the number of clusters is a key problem in data clustering. We present dip-means, a novel robust incremental method to learn the number of data clusters that may be used as a wrapper around any iterative clustering algorithm of the k-means family. In contrast to many popular methods which make assumptions about the underlying cluster distributions, dip-means only assumes a fundamental cluster property: each cluster to admit a unimodal distribution. The proposed algorithm considers each cluster member as a ''viewer'' and applies a univariate statistic hypothesis test for unimodality (dip-test) on the distribution of the distances between the viewer and the cluster members. Two important advantages are: i) the unimodality test is applied on univariate distance vectors, ii) it can be directly applied with kernel-based methods, since only the pairwise distances are involved in the computations.

Links between probabilistic and non-probabilistic learning algorithms can arise by performing small-variance asymptotics, i.e., letting the variance of particular distributions in a graphical model go to zero. For instance, in the context of clustering, such an approach yields precise connections between the k-means and EM algorithms. In this paper, we explore small-variance asymptotics for exponential family Dirichlet process (DP) and hierarchical Dirichlet process (HDP) mixture models. Utilizing connections between exponential family distributions and Bregman divergences, we derive novel clustering algorithms from the asymptotic limit of the DP and HDP mixtures that feature the scalability of existing hard clustering methods as well as the flexibility of Bayesian nonparametric models. We focus on special cases of our analysis for discrete-data problems, including topic modeling, and we demonstrate the utility of our results by applying variants of our algorithms to problems arising in vision and document analysis.

Discovering hierarchical regularities in data is a key problem in interacting with large datasets, modeling cognition, and encoding knowledge. A previous Bayesian solution---Kingman's coalescent---provides a convenient probabilistic model for data represented as a binary tree. Unfortunately, this is inappropriate for data better described by bushier trees. We generalize an existing belief propagation framework of Kingman's coalescent to the beta coalescent, which models a wider range of tree structures. Because of the complex combinatorial search over possible structures, we develop new sampling schemes using sequential Monte Carlo and Dirichlet process mixture models, which render inference efficient and tractable.

The introduction of embedding techniques has pushed forward significantly the Natural Language Processing field. Many of the proposed solutions have been presented for word-level encoding; anyhow, in the last years, new mechanism to treat information at an higher level of aggregation, like at sentence- and document-level, have emerged. With this work we address specifically the sentence embeddings problem, presenting the Static Fuzzy Bag-of-Word model. Our model is a refinement of the Fuzzy Bag-of-Words approach, providing sentence embeddings with a predefined dimension. SFBoW provides competitive performances in Semantic Textual Similarity benchmarks, while requiring low computational resources.

Data-Driven Response Regime Exploration and Identification (DR$^2$EI) is a novel and fully data-driven method for identifying and classifying response regimes of a dynamical system without requiring human intervention. This approach is a valuable tool for exploring and discovering response regimes in complex dynamical systems, especially when the governing equations and the number of response regimes are unknown, and the system is expensive to sample. Additionally, the method is useful for order reduction, as it can be used to identify the most dominant response regimes of a given dynamical system. DR$^2$EI utilizes unsupervised learning algorithms to transform the system's response into an embedding space that facilitates regime classification. An active sequential sampling approach based on Gaussian Process Regression (GPR) is used to efficiently sample the parameter space, quantify uncertainty, and provide optimal trade-offs between exploration and exploitation. The performance of the DR$^2$EI method was evaluated by analyzing three established dynamical systems: the mathematical pendulum, the Lorenz system, and the Duffing oscillator. The method was shown to effectively identify a variety of response regimes with both similar and distinct topological features and frequency content, demonstrating its versatility in capturing a wide range of behaviors. While it may not be possible to guarantee that all possible regimes will be identified, the method provides an automated and efficient means for exploring the parameter space of a dynamical system and identifying its underlying "sufficiently dominant" response regimes without prior knowledge of the system's equations or behavior.

This paper considers the well-studied algorithmic regime of designing a $(1+\epsilon)$-approximation algorithm for a $k$-clustering problem that runs in time $f(k,\epsilon)poly(n)$ (sometimes called an efficient parameterized approximation scheme or EPAS for short). Notable results of this kind include EPASes in the high-dimensional Euclidean setting for $k$-center [Bad\u{o}iu, Har-Peled, Indyk; STOC'02] as well as $k$-median, and $k$-means [Kumar, Sabharwal, Sen; J. ACM 2010]. However, existing EPASes handle only basic objectives (such as $k$-center, $k$-median, and $k$-means) and are tailored to the specific objective and metric space. Our main contribution is a clean and simple EPAS that settles more than ten clustering problems (across multiple well-studied objectives as well as metric spaces) and unifies well-known EPASes. Our algorithm gives EPASes for a large variety of clustering objectives (for example, $k$-means, $k$-center, $k$-median, priority $k$-center, $\ell$-centrum, ordered $k$-median, socially fair $k$-median aka robust $k$-median, or more generally monotone norm $k$-clustering) and metric spaces (for example, continuous high-dimensional Euclidean spaces, metrics of bounded doubling dimension, bounded treewidth metrics, and planar metrics). Key to our approach is a new concept that we call bounded $\epsilon$-scatter dimension--an intrinsic complexity measure of a metric space that is a relaxation of the standard notion of bounded doubling dimension. Our main technical result shows that two conditions are essentially sufficient for our algorithm to yield an EPAS on the input metric $M$ for any clustering objective: (i) The objective is described by a monotone (not necessarily symmetric!) norm, and (ii) the $\epsilon$-scatter dimension of $M$ is upper bounded by a function of $\epsilon$.

Spectral Toolkit of Algorithms for Graphs (STAG) is an open-source C++ and Python library of efficient spectral algorithms for graphs. Our objective is to implement advanced graph algorithms developed through algorithmic spectral graph theory, while making it practical to end users. This series of technical reports is to document our progress on STAG, including implementation details, engineering considerations, and the data sets against which our implementation is tested. The report is structured as follows: Section 2 describes the local clustering algorithm, which is the main update in this STAG release. The discussion is at a high level such that domain knowledge beyond basic algorithms is not needed. Section 3 provides a user guide to the essential features of STAG which allow a user to apply local clustering. Section 4 includes experiments and demonstrations of the functionality of STAG. Finally, Section 5 discusses several technical details; these include our choice of implemented algorithms, the default setup of parameters, and other technical choices. We leave these details to the final section, as it's not necessary for the reader to understand this when using STAG.

The interactive exploration of large and evolving datasets is challenging as relationships between underlying variables may not be fully understood. There may be hidden trends and patterns in the data that are worthy of further exploration and analysis. We present a system that methodically explores multiple combinations of variables using a searchlight technique and identifies outliers. An iterative k-means clustering algorithm is applied to features derived through a split-apply-combine paradigm used in the database literature. Outliers are identified as singleton or small clusters. This algorithm is swept across the dataset in a searchlight manner. The dimensions that contain outliers are combined in pairs with other dimensions using a susbset scan technique to gain further insight into the outliers. We illustrate this system by anaylzing open health care data released by New York State. We apply our iterative k-means searchlight followed by subset scanning. Several anomalous trends in the data are identified, including cost overruns at specific hospitals, and increases in diagnoses such as suicides. These constitute novel findings in the literature, and are of potential use to regulatory agencies, policy makers and concerned citizens.

With calls for increasing transparency, governments are releasing greater amounts of data in multiple domains including finance, education and healthcare. The efficient exploratory analysis of healthcare data constitutes a significant challenge. Key concerns in public health include the quick identification and analysis of trends, and the detection of outliers. This allows policies to be rapidly adapted to changing circumstances. We present an efficient outlier detection technique, termed PIKS (Pruned iterative-k means searchlight), which combines an iterative k-means algorithm with a pruned searchlight based scan. We apply this technique to identify outliers in two publicly available healthcare datasets from the New York Statewide Planning and Research Cooperative System, and California's Office of Statewide Health Planning and Development. We provide a comparison of our technique with three other existing outlier detection techniques, consisting of auto-encoders, isolation forests and feature bagging. We identified outliers in conditions including suicide rates, immunity disorders, social admissions, cardiomyopathies, and pregnancy in the third trimester. We demonstrate that the PIKS technique produces results consistent with other techniques such as the auto-encoder. However, the auto-encoder needs to be trained, which requires several parameters to be tuned. In comparison, the PIKS technique has far fewer parameters to tune. This makes it advantageous for fast, "out-of-the-box" data exploration. The PIKS technique is scalable and can readily ingest new datasets. Hence, it can provide valuable, up-to-date insights to citizens, patients and policy-makers. We have made our code open source, and with the availability of open data, other researchers can easily reproduce and extend our work. This will help promote a deeper understanding of healthcare policies and public health issues.