Random column sampling is not guaranteed to yield data sketches that preserve the underlying structures of the data and may not sample sufficiently from less-populated data clusters. Also, adaptive sampling can often provide accurate low rank approximations, yet may fall short of producing descriptive data sketches, especially when the cluster centers are linearly dependent. Motivated by that, this paper introduces a novel randomized column sampling tool dubbed Spatial Random Sampling (SRS), in which data points are sampled based on their proximity to randomly sampled points on the unit sphere. The most compelling feature of SRS is that the corresponding probability of sampling from a given data cluster is proportional to the surface area the cluster occupies on the unit sphere, independently from the size of the cluster population. Although it is fully randomized, SRS is shown to provide descriptive and balanced data representations. The proposed idea addresses a pressing need in data science and holds potential to inspire many novel approaches for analysis of big data.

This paper presents a remarkably simple, yet powerful, algorithm termed Coherence Pursuit (CoP) to robust Principal Component Analysis (PCA). As inliers lie in a low dimensional subspace and are mostly correlated, an inlier is likely to have strong mutual coherence with a large number of data points. By contrast, outliers either do not admit low dimensional structures or form small clusters. In either case, an outlier is unlikely to bear strong resemblance to a large number of data points. Given that, CoP sets an outlier apart from an inlier by comparing their coherence with the rest of the data points. The mutual coherences are computed by forming the Gram matrix of the normalized data points. Subsequently, the sought subspace is recovered from the span of the subset of the data points that exhibit strong coherence with the rest of the data. As CoP only involves one simple matrix multiplication, it is significantly faster than the state-of-the-art robust PCA algorithms. We derive analytical performance guarantees for CoP under different models for the distributions of inliers and outliers in both noise-free and noisy settings. CoP is the first robust PCA algorithm that is simultaneously non-iterative, provably robust to both unstructured and structured outliers, and can tolerate a large number of unstructured outliers.

In this paper we propose a mixture model, SparseMix, for clustering of sparse high dimensional binary data, which connects model-based with centroid-based clustering. Every group is described by a representative and a probability distribution modeling dispersion from this representative. In contrast to classical mixture models based on EM algorithm, SparseMix: -is especially designed for the processing of sparse data, -can be efficiently realized by an on-line Hartigan optimization algorithm, -is able to automatically reduce unnecessary clusters. We perform extensive experimental studies on various types of data, which confirm that SparseMix builds partitions with higher compatibility with reference grouping than related methods. Moreover, constructed representatives often better reveal the internal structure of data.

We develop a model in which interactions between nodes of a dynamic network are counted by non homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then a maximum-likelihood estimator is developed to estimate non parametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology.

The stochastic block model (SBM) is a flexible probabilistic tool that can be used to model interactions between clusters of nodes in a network. However, it does not account for interactions of time varying intensity between clusters. The extension of the SBM developed in this paper addresses this shortcoming through a temporal partition: assuming interactions between nodes are recorded on fixed-length time intervals, the inference procedure associated with the model we propose allows to cluster simultaneously the nodes of the network and the time intervals. The number of clusters of nodes and of time intervals, as well as the memberships to clusters, are obtained by maximizing an exact integrated complete-data likelihood, relying on a greedy search approach. Experiments on simulated and real data are carried out in order to assess the proposed methodology.

About • subfield of Artificial Intelligence (AI) • name is derived from the concept that it deals with "construction and study of systems that can learn from data" • can be seen as building blocks to make computers learn to behave more intelligently • It is a theoretical concept. There are various techniques with various implementations.

In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of $N$ points in $d$ dimensions generated by an unknown mixture of $k$ spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version of the popular Lloyd's heuristic and show that the algorithm estimates all the unknown centers of the component Gaussians accurately if they are sufficiently separated. Assuming each pair of centers are $C\sigma$ distant with $C=\Omega((k\log k)^{1/4}\sigma)$ and where $\sigma^2$ is the maximum variance of any Gaussian component, we show that asymptotically the algorithm estimates the centers optimally (up to constants); our center separation requirement matches the best known result for spherical Gaussians \citep{vempalawang}. For finite samples, we show that a bias term based on the initial estimate decreases at $O(1/{\rm poly}(N))$ rate while variance decreases at nearly optimal rate of $\sigma^2 d/N$. Our analysis requires seeding the algorithm with a good initial estimate of the true cluster centers for which we provide an online PCA based clustering algorithm. Indeed, the asymptotic per-step time complexity of our algorithm is the optimal $d\cdot k$ while space complexity of our algorithm is $O(dk\log k)$. In addition to the bias and variance terms which tend to $0$, the hard-thresholding based updates of streaming Lloyd's algorithm is agnostic to the data distribution and hence incurs an approximation error that cannot be avoided. However, by using a streaming version of the classical (soft-thresholding-based) EM method that exploits the Gaussian distribution explicitly, we show that for a mixture of two Gaussians the true means can be estimated consistently, with estimation error decreasing at nearly optimal rate, and tending to $0$ for $N\rightarrow \infty$.

A k-means clustering algorithmus implemented in p5.js It shows the clustering algorithm quite nicely in a visual way. I also liked p5.js a lot. The only thing which is neede is some kind of webserver to serve index.html, Click on the graphics to see how the points get clustered.

Extracting meaningful knowledge from large and nonlinearly-connected data structures is of primary importance for efficiently utilizing data. Big data problems (e.g. 1 GB/s) often contain superpositions of multiple distinct processes, sources, or latent factors. Estimating or inferring the component distributions or statistical factors is called the mixture problem. Methods for solving mixture problems are known as mixture models [Everitt, 1996], and in machine learning they are used to define Bayes classifiers [Bishop, 2006]. Mixture models are a widely applicable pattern recognition and dimensionality reduction approach for extracting meaningful content from large and complex datasets. Only finite mixture models are described here, although countably or uncountably infinite numbers of mixture components are also possible [McAuliffe et al., 2006]. In terms of dimensionality reduction methods, Laplacian mixture models provide global and nonhierarchical analyses of massive datasets using scalable algorithms.

As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of outliers. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure.