5,305 research outputs found
Foundational principles for large scale inference: Illustrations through correlation mining
When can reliable inference be drawn in the "Big Data" context? This paper
presents a framework for answering this fundamental question in the context of
correlation mining, with implications for general large scale inference. In
large scale data applications like genomics, connectomics, and eco-informatics
the dataset is often variable-rich but sample-starved: a regime where the
number of acquired samples (statistical replicates) is far fewer than the
number of observed variables (genes, neurons, voxels, or chemical
constituents). Much of recent work has focused on understanding the
computational complexity of proposed methods for "Big Data." Sample complexity
however has received relatively less attention, especially in the setting when
the sample size is fixed, and the dimension grows without bound. To
address this gap, we develop a unified statistical framework that explicitly
quantifies the sample complexity of various inferential tasks. Sampling regimes
can be divided into several categories: 1) the classical asymptotic regime
where the variable dimension is fixed and the sample size goes to infinity; 2)
the mixed asymptotic regime where both variable dimension and sample size go to
infinity at comparable rates; 3) the purely high dimensional asymptotic regime
where the variable dimension goes to infinity and the sample size is fixed.
Each regime has its niche but only the latter regime applies to exa-scale data
dimension. We illustrate this high dimensional framework for the problem of
correlation mining, where it is the matrix of pairwise and partial correlations
among the variables that are of interest. We demonstrate various regimes of
correlation mining based on the unifying perspective of high dimensional
learning rates and sample complexity for different structured covariance models
and different inference tasks
Scalable sparse covariance estimation via self-concordance
We consider the class of convex minimization problems, composed of a
self-concordant function, such as the metric, a convex data fidelity
term and, a regularizing -- possibly non-smooth -- function
. This type of problems have recently attracted a great deal of
interest, mainly due to their omnipresence in top-notch applications. Under
this \emph{locally} Lipschitz continuous gradient setting, we analyze the
convergence behavior of proximal Newton schemes with the added twist of a
probable presence of inexact evaluations. We prove attractive convergence rate
guarantees and enhance state-of-the-art optimization schemes to accommodate
such developments. Experimental results on sparse covariance estimation show
the merits of our algorithm, both in terms of recovery efficiency and
complexity.Comment: 7 pages, 1 figure, Accepted at AAAI-1
The Gaussian rank correlation estimator: Robustness properties.
The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix based on the Gaussian rank correlation is always positive semidefinite, and very easy to compute, also in high dimensions. A simulation study confirms the good efficiency and robustness properties of the proposed estimator with respect to the popular Kendall and Spearman correlation measures. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.Breakdown; Correlation; Efficiency; Robustness; Van der Waerden;
Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data
Subsequence clustering of multivariate time series is a useful tool for
discovering repeated patterns in temporal data. Once these patterns have been
discovered, seemingly complicated datasets can be interpreted as a temporal
sequence of only a small number of states, or clusters. For example, raw sensor
data from a fitness-tracking application can be expressed as a timeline of a
select few actions (i.e., walking, sitting, running). However, discovering
these patterns is challenging because it requires simultaneous segmentation and
clustering of the time series. Furthermore, interpreting the resulting clusters
is difficult, especially when the data is high-dimensional. Here we propose a
new method of model-based clustering, which we call Toeplitz Inverse
Covariance-based Clustering (TICC). Each cluster in the TICC method is defined
by a correlation network, or Markov random field (MRF), characterizing the
interdependencies between different observations in a typical subsequence of
that cluster. Based on this graphical representation, TICC simultaneously
segments and clusters the time series data. We solve the TICC problem through
alternating minimization, using a variation of the expectation maximization
(EM) algorithm. We derive closed-form solutions to efficiently solve the two
resulting subproblems in a scalable way, through dynamic programming and the
alternating direction method of multipliers (ADMM), respectively. We validate
our approach by comparing TICC to several state-of-the-art baselines in a
series of synthetic experiments, and we then demonstrate on an automobile
sensor dataset how TICC can be used to learn interpretable clusters in
real-world scenarios.Comment: This revised version fixes two small typos in the published versio
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