In an era of ubiquitous large-scale streaming data, the availability of data far exceeds the capacity of expert human analysts. In many settings, such data is either discarded or stored unprocessed in datacenters. This paper proposes a method of online data thinning, in which large-scale streaming datasets are winnowed to preserve unique, anomalous, or salient elements for timely expert analysis. At the heart of this proposed approach is an online anomaly detection method based on dynamic, low-rank Gaussian mixture models. Specifically, the high-dimensional covariances matrices associated with the Gaussian components are associated with low-rank models. According to this model, most observations lie near a union of subspaces. The low-rank modeling mitigates the curse of dimensionality associated with anomaly detection for high-dimensional data, and recent advances in subspace clustering and subspace tracking allow the proposed method to adapt to dynamic environments. Furthermore, the proposed method allows subsampling, is robust to missing data, and uses a mini-batch online optimization approach. The resulting algorithms are scalable, efficient, and are capable of operating in real time. Experiments on wide-area motion imagery and e-mail databases illustrate the efficacy of the proposed approach.
Subspace tracking is a classical problem that experienced recent attention with the development of algorithms that are robust to missing and outlier elements of the data points @math . For example, the Grassmannian Rank-One Update Subspace Estimation (GROUSE) @cite_86 , Parallel Estimation and Tracking by REcursive Least Squares (PETRELS) @cite_13 @cite_64 , and Robust Online Subspace Estimation and Tracking Algorithm (ROSETA) @cite_27 effectively track a single subspace using incomplete data vectors. These algorithms are capable of tracking and adapting to changing environments. The subspace model used in these methods, however, is inherently strong, whereas a plethora of empirical studies have demonstrated that high-dimensional data often lie near manifolds with non-negligible curvature @cite_75 @cite_51 @cite_68 .
@cite_64 @cite_75 @cite_27 @cite_51 @cite_86 @cite_68 @cite_13
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We consider the problem of reconstructing a data stream from a small subset of its entries, where the data stream is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. It is also important to track the change of underlying subspace for many applications. This problem can be viewed as a sequential low-rank matrix completion problem in which the subspace is learned in an online fashion. The proposed algorithm, called Parallel Estimation and Tracking by REcursive Least Squares (PETRELS), identifies the underlying low-dimensional subspace via a recursive procedure for each row of the subspace matrix in parallel, and then reconstructs the missing entries via least-squares estimation if required. PETRELS outperforms previous approaches by discounting observations in order to capture long-term behavior of the data stream and be able to adapt to it. Numerical examples are provided for direction-of-arrival estimation and matrix completion, comparing PETRELS with state of the art batch algorithms. Data sets are often modeled as samples from a probability distribution in RD, for D large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a d-dimensional manifold M, with d much smaller than D. When M is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of d vectors in RD (for example found by SVD), at a cost (n+D)d for n data points. When M is nonlinear, there are no “explicit” and algorithmically efficient constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box global optimization. In this paper we construct data-dependent multi-scale dictionaries that aim at efficiently encoding and manipulating the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa, in contrast with L1-type sparsity-seeking algorithms, but like adaptive nonlinear approximation in classical multi-scale analysis. In addition, data points are guaranteed to have a compressible representation in terms of the dictionary, depending on the assumptions on the geometry of the underlying probability distribution. In this paper, we present a robust online subspace estimation and tracking algorithm (ROSETA) that is capable of identifying and tracking a time-varying low dimensional subspace from incomplete measurements and in the presence of sparse outliers. Our algorithm minimizes a robust l 1 norm cost function between the observed measurements and their projection onto the estimated subspace. The projection coefficients and sparse outliers are computed using ADMM solver and the subspace estimate is updated using a proximal point iteration with adaptive parameter selection. We demonstrate using simulated experiments and a video background subtraction example that ROSETA succeeds in identifying and tracking low dimensional subspaces using fewer iterations than other state of art algorithms. Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text. How do we judge similarity? Our mental representations of the world are formed by processing large numbers of sensory in One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed. Many real world datasets exhibit an embedding of low-dimensional structure in a high-dimensional manifold. Examples include images, videos and internet traffic data. It is of great significance to estimate and track the low-dimensional structure with small storage requirements and computational complexity when the data dimension is high. Therefore we consider the problem of reconstructing a data stream from a small subset of its entries, where the data is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. We further consider tracking the change of the underlying subspace, which can be applied to applications such as video denoising, network monitoring and anomaly detection. Our setting can be viewed as a sequential low-rank matrix completion problem in which the subspace is learned in an online fashion. The proposed algorithm, dubbed Parallel Estimation and Tracking by REcursive Least Squares (PETRELS), first identifies the underlying low-dimensional subspace, and then reconstructs the missing entries via least-squares estimation if required. Subspace identification is performed via a recursive procedure for each row of the subspace matrix in parallel with discounting for previous observations. Numerical examples are provided for direction-of-arrival estimation and matrix completion, comparing PETRELS with state of the art batch algorithms.