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Spectral methods are of fundamental importance in statistics and machine learning, because they underlie algorithms from classical principal components analysis to more recent approaches that exploit manifold structure. In most cases, the core technical problem can be reduced to computing a low-rank approximation to a positive-definite kernel. For the growing number of applications dealing with very large or high-dimensional datasets, however, the optimal approximation afforded by an exact spectral decomposition is too costly, because its complexity scales as the cube of either the number of training examples or their dimensionality. Motivated by such applications, we present here 2 new algorithms for the approximation of positive-semidefinite kernels, together with error bounds that improve on results in the literature. We approach this problem by seeking to determine, in an efficient manner, the most informative subset of our data relative to the kernel approximation task at hand. This leads to two new strategies based on the Nystrom method that are directly applicable to massive datasets. The first of these--based on sampling--leads to a randomized algorithm whereupon the kernel induces a probability distribution on its set of partitions, whereas the latter approach--based on sorting--provides for the selection of a partition in a deterministic way. We detail their numerical implementation and provide simulation results for a variety of representative problems in statistical data analysis, each of which demonstrates the improved performance of our approach relative to existing methods. statistical data analysis | kernel methods | low-rank approximation

In this paper we introduce the network histogram, a statistical summary of network interactions to be used as a tool for exploratory data analysis. A network histogram is obtained by fitting a stochastic blockmodel to a single observation of a network dataset. Blocks of edges play the role of histogram bins and community sizes that of histogram bandwidths or bin sizes. Just as standard histograms allow for varying bandwidths, different blockmodel estimates can all be considered valid representations of an underlying probability model, subject to bandwidth constraints. Here we provide methods for automatic bandwidth selection, by which the network histogram approximates the generating mechanism that gives rise to exchangeable random graphs. This makes the blockmodel a universal network representation for unlabeled graphs. With this insight, we discuss the interpretation of network communities in light of the fact that many different community assignments can all give an equally valid representation of such a network. To demonstrate the fidelity-versus-interpretability tradeoff inherent in considering different numbers and sizes of communities, we analyze two publicly available networks--political weblogs and student friendships--and discuss how to interpret the network histogram when additional information related to node and edge labeling is present. [ABSTRACT FROM AUTHOR]

The author discusses mathematical methods for simplification of big data. The author says that Parson has utilizes Gauss' method of least squares to solve equations of high-dimensional dataset. He states that multidimentional scaling used for data dimension reduction portrays computational cost which is dominated by dimentionality. Moreover, he mentions that mathematical methods for data analysis is important for transformation of outrageous dataset into an asset.