Spectral methods have gained a lot of recent attention due to the simplicity of their implementation and their solid mathematical background. We revisit spectral graph clustering, and reformulate in the $p$-norm the continuous problem of minimizing the graph Laplacian Rayleigh quotient. The value of $p \in (1,2]$ is reduced, promoting sparser solution vectors that correspond to optimal clusters as $p$ approaches one. The computation of multiple $p$-eigenvectors of the graph $p$-Laplacian, a nonlinear generalization of the standard graph Laplacian, is achieved by the minimization of our objective function on the Grassmann manifold, hence ensuring the enforcement of the orthogonality constraint between them. Our approach attempts to bridge the fields of graph clustering and nonlinear numerical optimization, and employs a robust algorithm to obtain clusters of high quality. The benefits of the suggested method are demonstrated in a plethora of artificial and real-world graphs. Our results are compared against standard spectral clustering methods and the current state-of-the-art algorithm for clustering using the graph $p$-Laplacian variant.

Response process data collected from human-computer interactive items contain rich information about respondents' behavioral patterns and cognitive processes. Their irregular formats as well as their large sizes make standard statistical tools difficult to apply. This paper develops a computationally efficient method for exploratory analysis of such process data. The new approach segments a lengthy individual process into a sequence of short subprocesses to achieve complexity reduction, easy clustering and meaningful interpretation. Each subprocess is considered a subtask. The segmentation is based on sequential action predictability using a parsimonious predictive model combined with the Shannon entropy. Simulation studies are conducted to assess performance of the new methods. We use the process data from PIAAC 2012 to demonstrate how exploratory analysis of process data can be done with the new approach.

Deciding on the unimodality of a dataset is an important problem in data analysis and statistical modeling. It allows to obtain knowledge about the structure of the dataset, ie. whether data points have been generated by a probability distribution with a single or more than one peaks. Such knowledge is very useful for several data analysis problems, such as for deciding on the number of clusters and determining unimodal projections. We propose a technique called UU-test (Unimodal Uniform test) to decide on the unimodality of a one-dimensional dataset. The method operates on the empirical cumulative density function (ecdf) of the dataset. It attempts to build a piecewise linear approximation of the ecdf that is unimodal and models the data sufficiently in the sense that the data corresponding to each linear segment follows the uniform distribution. A unique feature of this approach is that in the case of unimodality, it also provides a statistical model of the data in the form of a Uniform Mixture Model. We present experimental results in order to assess the ability of the method to decide on unimodality and perform comparisons with the well-known dip-test approach. In addition, in the case of unimodal datasets we evaluate the Uniform Mixture Models provided by the proposed method using the test set log-likelihood and the two-sample Kolmogorov-Smirnov (KS) test.

This paper studies the statistical and computational limits of high-order clustering with planted structures. We focus on two clustering models, constant high-order clustering (CHC) and rank-one higher-order clustering (ROHC), and study the methods and theory for testing whether a cluster exists (detection) and identifying the support of cluster (recovery). Specifically, we identify the sharp boundaries of signal-to-noise ratio for which CHC and ROHC detection/recovery are statistically possible. We also develop the tight computational thresholds: when the signal-to-noise ratio is below these thresholds, we prove that polynomial-time algorithms cannot solve these problems under the computational hardness conjectures of hypergraphic planted clique (HPC) detection and hypergraphic planted dense subgraph (HPDS) recovery. We also propose polynomial-time tensor algorithms that achieve reliable detection and recovery when the signal-to-noise ratio is above these thresholds. Both sparsity and tensor structures yield the computational barriers in high-order tensor clustering. The interplay between them results in significant differences between high-order tensor clustering and matrix clustering in literature in aspects of statistical and computational phase transition diagrams, algorithmic approaches, hardness conjecture, and proof techniques. To our best knowledge, we are the first to give a thorough characterization of the statistical and computational trade-off for such a double computational-barrier problem. Finally, we provide evidence for the computational hardness conjectures of HPC detection and HPDS recovery.

Determining the number of clusters that best partitions a dataset can be a challenging task because of 1) the lack of a priori information within an unsupervised learning framework; and 2) the absence of a unique clustering validation approach to evaluate clustering solutions. Here we present reval: a Python package that leverages stability-based relative clustering validation methods to determine best clustering solutions. Statistical software, both in R and Python, usually rely on internal validation metrics, such as the silhouette index, to select the number of clusters that best fits the data. Meanwhile, open-source software solutions that easily implement relative clustering techniques are lacking. Internal validation methods exploit characteristics of the data itself to produce a result, whereas relative approaches attempt to leverage the unknown underlying distribution of data points looking for a replicable and generalizable clustering solution. The implementation of relative validation solutions can further the theory of clustering by enriching the already available methods that can be used to investigate clustering results in different situations and for different data distributions. This work aims at contributing to this effort by developing a stability-based method that selects the best clustering solution as the one that replicates, via supervised learning, on unseen subsets of data. The package works with multiple clustering and classification algorithms, hence allowing further assessment of the stability of different clustering mechanisms.

K means clustering is another simplified algorithm in machine learning. It is categorized into unsupervised learning because here we don't know the result already (no idea about which cluster will be formed). This algorithm is used for vector quantization of the data and has been taken from signal processing methodology. Here the data is divided into several groups, data points in each group have similar characteristics. These clusters are decided by calculating the distance between data points.

Multi-component polymer systems are of interest in organic photovoltaic and drug delivery applications, among others where diverse morphologies influence performance. An improved understanding of morphology classification, driven by composition-informed prediction tools, will aid polymer engineering practice. We use a modified Cahn-Hilliard model to simulate polymer precipitation. Such physics-based models require high-performance computations that prevent rapid prototyping and iteration in engineering settings. To reduce the required computational costs, we apply machine learning techniques for clustering and consequent prediction of the simulated polymer blend images in conjunction with simulations. Integrating ML and simulations in such a manner reduces the number of simulations needed to map out the morphology of polymer blends as a function of input parameters and also generates a data set which can be used by others to this end. We explore dimensionality reduction, via principal component analysis and autoencoder techniques, and analyse the resulting morphology clusters. Supervised machine learning using Gaussian process classification was subsequently used to predict morphology clusters according to species molar fraction and interaction parameter inputs. Manual pattern clustering yielded the best results, but machine learning techniques were able to predict the morphology of polymer blends with $\geq$ 90 $\%$ accuracy.

Industry 4.0 is a concept which helps companies to have a smart supply chain system when they are faced with a dynamic process. As Industry 4.0 focuses on mobility and real-time integration, it is a good framework for a Dynamic Vehicle Routing problem (DVRP). The main objective of this research is to solve the DVRP on a large-scale size. The aim of this study is to show that the delivery vehicles must serve customer demands from a common depot to have a minimum transit cost without exceeding the capacity constraint of each vehicle. In VRP, to reach an exact solution is quite difficult, and in large-size real world problems it is often impossible. Also, the computational time complexity of this type of problem grows exponentially. In order to find optimal answers for this problem in medium and large dimensions, using a heuristic approach is recommended as the best approach. A hierarchical approach consisting of three stages as cluster-first, route-construction second, route-improvement third is proposed. In the first stage, customers are clustered based on the number of vehicles with different clustering algorithms (i.e., K-mean, GMM, and BIRCH algorithms). In the second stage, the DVRP is solved using construction algorithms and in the third stage improvement algorithms are applied. The second stage is solved using construction algorithms (i.e. Savings algorithm, path cheapest arc algorithm, etc.). In the third stage, improvement algorithms such as Guided Local Search, Simulated Annealing and Tabu Search are applied. One of the main contributions of this paper is that the proposed approach can deal with large-size real world problems to decrease the computational time complexity. The results of this approach confirmed that the proposed methodology is applicable.

Say you've got a dataset where there exist relationships between individual samples, and your goal is to identify groups of related samples within the dataset. Clustering, which is part of the class of unsupervised machine learning algorithms, is then the way to go. But what clustering algorithm to apply when you do not really know the number of clusters? Enter Affinity Propagation, a gossip-style algorithm which derives the number of clusters by mimicing social group formation by passing messages about the popularity of individual samples as to whether they're part of a certain group, or even if they are the leader of one. This algorithm, which can estimate the number of clusters/groups in your dataset itself, is the topic of today's blog post.

The ocean is filled with microscopic microalgae called phytoplankton, which together are responsible for as much photosynthesis as all plants on land combined. Our ability to predict their response to the warming ocean relies on understanding how the dynamics of phytoplankton populations is influenced by changes in environmental conditions. One powerful technique to study the dynamics of phytoplankton is flow cytometry, which measures the optical properties of thousands of individual cells per second. Today, oceanographers are able to collect flow cytometry data in real-time onboard a moving ship, providing them with fine-scale resolution of the distribution of phytoplankton across thousands of kilometers. One of the current challenges is to understand how these small and large scale variations relate to environmental conditions, such as nutrient availability, temperature, light and ocean currents. In this paper, we propose a novel sparse mixture of multivariate regressions model to estimate the time-varying phytoplankton subpopulations while simultaneously identifying the specific environmental covariates that are predictive of the observed changes to these subpopulations. We demonstrate the usefulness and interpretability of the approach using both synthetic data and real observations collected on an oceanographic cruise conducted in the north-east Pacific in the spring of 2017.