The concept of sample mean in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems and generalize centroid-based clustering algorithms. Its existence has neither been proved nor challenged. This article presents sufficient conditions for existence of a sample mean in DTW spaces. The proposed result justifies prior work on approximate mean algorithms, sets the stage for constructing exact mean algorithms, and is a first step towards a statistical theory of DTW spaces.

The Power Mean Laplacian for Multilayer Graph ClusteringPedro Mercado 1 Antoine Gautier 1 Francesco Tudisco 2 Matthias Hein 1 1 Department of Mathematics and Computer Science, Saarland University, Germany 2 Department of Mathematics and Statistics, University of Strathclyde, G11XH Glasgow, UK Abstract Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information from different layers. We introduce in this paper a one-parameter family of matrix power means for merging the Laplacians from different layers and analyze it in expectation in the stochastic block model. We show that this family allows to recover ground truth clusters under different settings and verify this in real world data. While computing the matrix power mean can be very expensive for large graphs, we introduce a numerical scheme to efficiently compute its eigenvectors for the ...

The problem of complex data analysis is a central topic of modern statistical science and learning systems and is becoming of broader interest with the increasing prevalence of high-dimensional data. The challenge is to develop statistical models and autonomous algorithms that are able to acquire knowledge from raw data for exploratory analysis, which can be achieved through clustering techniques or to make predictions of future data via classification (i.e., discriminant analysis) techniques. Latent data models, including mixture model-based approaches are one of the most popular and successful approaches in both the unsupervised context (i.e., clustering) and the supervised one (i.e, classification or discrimination). Although traditionally tools of multivariate analysis, they are growing in popularity when considered in the framework of functional data analysis (FDA). FDA is the data analysis paradigm in which the individual data units are functions (e.g., curves, surfaces), rather than simple vectors. In many areas of application, the analyzed data are indeed often available in the form of discretized values of functions or curves (e.g., time series, waveforms) and surfaces (e.g., 2d-images, spatio-temporal data). This functional aspect of the data adds additional difficulties compared to the case of a classical multivariate (non-functional) data analysis. We review and present approaches for model-based clustering and classification of functional data. We derive well-established statistical models along with efficient algorithmic tools to address problems regarding the clustering and the classification of these high-dimensional data, including their heterogeneity, missing information, and dynamical hidden structure. The presented models and algorithms are illustrated on real-world functional data analysis problems from several application area.

Vector quantization aims to form new vectors/matrices with shared values close to the original. It could compress data with acceptable information loss, and could be of great usefulness in areas like Image Processing, Pattern Recognition and Machine Learning. In recent years, the importance of quantization has been soaring as it has been discovered huge potentials in deploying practical neural networks, which is among one of the most popular research topics. Conventional vector quantization methods usually suffer from their own flaws: hand-coding domain rules quantization could produce poor results when encountering complex data, and clustering-based algorithms have the problem of inexact solution and high time consumption. In this paper, we explored vector quantization problem from a new perspective of sparse least square optimization and designed multiple algorithms with their program implementations. Specifically, deriving from a sparse form of coefficient matrix, three types of sparse least squares, with $l_0$, $l_1$, and generalized $l_1 + l_2$ penalizations, are designed and implemented respectively. In addition, to produce quantization results with given amount of quantized values(instead of penalization coefficient $\lambda$), this paper proposed a cluster-based least square quantization method, which could also be regarded as an improvement of information preservation of conventional clustering algorithm. The algorithms were tested on various data and tasks and their computational properties were analyzed. The paper offers a new perspective to probe the area of vector quantization, while the algorithms proposed could provide more appropriate options for quantization tasks under different circumstances.

Real-time advertising allows advertisers to bid for each impression for a visiting user. To optimize a specific goal such as maximizing the revenue led by ad placements, advertisers not only need to estimate the relevance between the ads and user's interests, but most importantly require a strategic response with respect to other advertisers bidding in the market. In this paper, we formulate bidding optimization with multi-agent reinforcement learning. To deal with a large number of advertisers, we propose a clustering method and assign each cluster with a strategic bidding agent. A practical Distributed Coordinated Multi-Agent Bidding (DCMAB) has been proposed and implemented to balance the tradeoff between the competition and cooperation among advertisers. The empirical study on our industry-scaled real-world data has demonstrated the effectiveness of our modeling methods. Our results show that a cluster based bidding would largely outperform single-agent and bandit approaches, and the coordinated bidding achieves better overall objectives than the purely self-interested bidding agents.

Words: 4252 Keywords: self-monitoring data, type 2 diabetes, machine learning, phenotyping, precision medicine ABSTRACT Objective: To evaluate unsupervised clustering methods for identifying individual-level behavioral-clinical phenotypes that relate personal biomarkers and behavioral traits in type 2 diabetes (T2DM) self-monitoring data. Materials and Methods: We used hierarchical clustering (HC) to identify groups of meals with similar nutrition and glycemic impact for 6 individuals with T2DM who collected self-monitoring data. We evaluated clusters on: 1) correspondence to gold standards generated by certified diabetes educators (CDEs) for 3 participants; 2) face validity, rated by CDEs, and 3) impact on CDEs' ability to identify patterns for another 3 participants. Results: Gold standard (GS) included 9 patterns across 3 participants. Of these, all 9 were rediscovered using HC: 4 GS patterns were consistent with patterns identified by HC (over 50% of meals in a cluster followed the pattern); another 5 were included as subgroups in broader clusers. After reviewing clusters, CDEs identified patterns that were more consistent with data (70% reduction in contradictions between patterns and participants' records). Discussion: Hierarchical clustering of blood glucose and macronutrient consumption appears suitable for discovering behavioral-clinical phenotypes in T2DM. Most clusters corresponded to gold standard and were rated positively by CDEs for face validity. Cluster visualizations helped CDEs identify more robust patterns in nutrition and glycemic impact, creating new possibilities for visual analytic solutions. Conclusion: Machine learning methods can use diabetes self-monitoring data to create personalized behavioral-clinical phenotypes, which may prove useful for delivering personalized medicine.

Sparse subspace clustering (SSC) is one of the current state-of-the-art methods for partitioning data points into the union of subspaces, with strong theoretical guarantees. However, it is not practical for large data sets as it requires solving a LASSO problem for each data point, where the number of variables in each LASSO problem is the number of data points. To improve the scalability of SSC, we propose to select a few sets of anchor points using a randomized hierarchical clustering method, and, for each set of anchor points, solve the LASSO problems for each data point allowing only anchor points to have a non-zero weight (this reduces drastically the number of variables). This generates a multilayer graph where each layer corresponds to a different set of anchor points. Using the Grassmann manifold of orthogonal matrices, the shared connectivity among the layers is summarized within a single subspace. Finally, we use $k$-means clustering within that subspace to cluster the data points, similarly as done by spectral clustering in SSC. We show on both synthetic and real-world data sets that the proposed method not only allows SSC to scale to large-scale data sets, but that it is also much more robust as it performs significantly better on noisy data and on data with close susbspaces and outliers, while it is not prone to oversegmentation.

The $k$-means algorithm is the most popular nonparametric clustering method in use, but cannot generally be applied to data sets with missing observations. The usual practice with such data sets is to either impute the values under an assumption of a missing-at-random mechanism or to ignore the incomplete records, and then to use the desired clustering method. We develop an efficient version of the $k$-means algorithm that allows for clustering cases where not all the features have observations recorded. Our extension is called $k_m$-means and reduces to the $k$-means algorithm when all records are complete. We also provide strategies to initialize our algorithm and to estimate the number of groups in the data set. Illustrations and simulations demonstrate the efficacy of our approach in a variety of settings and patterns of missing data. Our methods are also applied to the clustering of gamma-ray bursts and to the analysis of activation images obtained from a functional Magnetic Resonance Imaging experiment.

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification.

Scaling clustering algorithms to massive data sets is a challenging task. Recently, several successful approaches based on data summarization methods, such as coresets and sketches, were proposed. While these techniques provide provably good and small summaries, they are inherently problem dependent -- the practitioner has to commit to a fixed clustering objective before even exploring the data. However, can one construct small data summaries for a wide range of clustering problems simultaneously? We affirmatively answer this question by proposing an efficient algorithm that constructs such one-shot summaries for a large family of k-clustering problems while retaining strong theoretical guarantees.