We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation. A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of $(1-\delta)$ fraction of the points for any $\delta\in(0,1)$. As a special case, we show that under the $d$-dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as $\sqrt{1/d}$, which complements previous exact recovery results that require constant separation.

Manifold learning based methods have been widely used for non-linear dimensionality reduction (NLDR). However, in many practical settings, the need to process streaming data is a challenge for such methods, owing to the high computational complexity involved. Moreover, most methods operate under the assumption that the input data is sampled from a single manifold, embedded in a high dimensional space. We propose a method for streaming NLDR when the observed data is either sampled from multiple manifolds or irregularly sampled from a single manifold. We show that existing NLDR methods, such as Isomap, fail in such situations, primarily because they rely on smoothness and continuity of the underlying manifold, which is violated in the scenarios explored in this paper. However, the proposed algorithm is able to learn effectively in presence of multiple, and potentially intersecting, manifolds, while allowing for the input data to arrive as a massive stream.

Here, I've used the famous Iris Flower dataset to show the clustering in Power BI using R. I've used the K-means clustering method to show the different species of Iris flower. About the dataset: The Iris dataset has 5 attributes (Sepal length, Sepal width, Petal width, Petal length, Species). The 3 different species are named as Setosa, Versicolor and Virginica. It is observed that, the Petal Length and Petal Width are similar in each Species, hence I have considered Petal Length for x axis and Petal Width for y axis to plot a graph. K-means Clustering: K means is a non-hierarchical iterative clustering technique.In this technique we start by randomly assigning the data points to clusters.

Data quality issues have attracted widespread attention due to the negative impacts of dirty data on data mining and machine learning results. The relationship between data quality and the accuracy of results could be applied on the selection of the appropriate algorithm with the consideration of data quality and the determination of the data share to clean. However, rare research has focused on exploring such relationship. Motivated by this, this paper conducts an experimental comparison for the effects of missing, inconsistent and conflicting data on classification, clustering, and regression algorithms. Based on the experimental findings, we provide guidelines for algorithm selection and data cleaning.

Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficiency and scalability of traditional spectral clustering methods can not be guaranteed for large scale datasets. In this paper, we propose a subspace clustering model based on the Kronecker product. Due to the property that the Kronecker product of a block diagonal matrix with any other matrix is still a block diagonal matrix, we can efficiently learn the representation matrix which is formed by the Kronecker product of k smaller matrices. By doing so, our model significantly reduces the computational complexity to O(kN^{3/k}). Furthermore, our model is general in nature, and can be adapted to different regularization based subspace clustering methods. Experimental results on two public datasets show that our model significantly improves the efficiency compared with several state-of-the-art methods. Moreover, we have conducted experiments on synthetic data to verify the scalability of our model for large scale datasets.

We consider the analysis of spectral clustering algorithms for community detection under a stochastic block model (SBM). A general spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm in the reduced spectral domain. By varying each step, one can obtain different spectral algorithms. In light of the recent developments in refining consistency results for the spectral clustering, we identify the necessary bounds at each of these three steps, and then derive and compare consistency results for some existing spectral algorithms as well as a new variant that we propose. The focus of the paper is on providing a better understanding of the analysis of spectral methods for community detection, with an emphasis on the bipartite setting which has received less theoretical consideration. We show how the variations in the spectral truncation step reflects in the consistency results under a general SBM. We also investigate the necessary bounds for the k-means step in some detail, allowing one to replace this step with any algorithm (k-means type or otherwise) that guarantees the necessary bound. We discuss some of the neglected aspects of the bipartite setting, e.g., the role of the mismatch between the communities of the two sides on the performance of spectral methods. Finally, we show how the consistency results can be extended beyond SBMs to the problem of clustering inhomogeneous random graph models that can be approximated by SBMs in a certain sense.

When our data is relatively clean and low-dimensional, looking at a table of summary statistics or some scatter plots can usually reveal how good clustering would be on the data. Look for things like large'clumps' of points in scatter plots between features, large variances, large differences between median and mean, properties of data between quantiles etc.

Environmental monitoring allows authorities to understand the impact of potentially harmful phenomena, such as air pollution, excessive noise, and radiation. Recently, there has been considerable interest in participatory sensing as a paradigm for such large-scale data collection because it is cost-effective and able to capture more fine-grained data than traditional approaches that use stationary sensors scattered in cities. In this approach, ordinary citizens (non-expert contributors) collect environmental data using low-cost mobile devices. However, these participants are generally self-interested actors that have their own goals and make local decisions about when and where to take measurements. This can lead to highly inefficient outcomes, where observations are either taken redundantly or do not provide sufficient information about key areas of interest. To address these challenges, it is necessary to guide and to coordinate participants, so they take measurements when it is most informative. To this end, we develop a computationally-efficient coordination algorithm (adaptive Best-Match) that suggests to users when and where to take measurements. Our algorithm exploits probabilistic knowledge of human mobility patterns, but explicitly considers the uncertainty of these patterns and the potential unwillingness of people to take measurements when requested to do so. In particular, our algorithm uses a local search technique, clustering and random simulations to map participants to measurements that need to be taken in space and time. We empirically evaluate our algorithm on a real-world human mobility and air quality dataset and show that it outperforms the current state of the art by up to 24% in terms of utility gained.

Quantifying similarity between data objects is an important part of modern data science. Deciding what similarity measure to use is very application dependent. In this paper, we combine insights from systems theory and machine learning, and investigate the weighted cepstral distance, which was previously defined for signals coming from ARMA models. We provide an extension of this distance to invertible deterministic linear time invariant single input single output models, and assess its applicability. We show that it can always be interpreted in terms of the poles and zeros of the underlying model, and that, in the case of stable, minimum-phase, or unstable, maximum-phase models, a geometrical interpretation in terms of subspace angles can be given. We then devise a method to assess stability and phase-type of the generating models, using only input/output signal information. In this way, we prove a connection between the extended weighted cepstral distance and a weighted cepstral model norm. In this way, we provide a purely data-driven way to assess different underlying dynamics of input/output signal pairs, without the need for any system identification step. This can be useful in machine learning tasks such as time series clustering. An iPython tutorial is published complementary to this paper, containing implementations of the various methods and algorithms presented here, as well as some numerical illustrations of the equivalences proven here.

Dimensionality is an ever present concern with data becoming increasingly higher dimensional over the last few years. To combat this issue, dimension reduction techniques have become very important tools, especially in the area of clustering (unsupervised classification) as well as semi-supervised and supervised classification. For multivariate data, the mixture of factor analyzers model has proved to be very useful in this regard as the model performs clustering and dimension reduction simultaneously, details in Section 2. However, there is 1 relative paucity in the area of dimension reduction for use in model-based clustering for matrix variate data. Matrix variate distributions have been shown to be useful for modelling three way data such as images and multivariate longitudinal data; however, the methods presented in the literature do suffer from dimensionality concerns. In this paper, we present a mixture of matrix variate bilinear factor analyzers (MMVBFA) model for use in clustering for higher dimensional matrix data. The matrix variate bilinear factor analyzers model can be viewed as a generalization of bilinear principal component analysis (BPCA; Zhao et al. 2012), and contains BPCA as a special case. An alternating expectation conditional maximization (AECM) algorithm (Meng & van Dyk 1997) is used for parameter estimation. The proposed method is illustrated using both simulated and real datasets.