88,703 research outputs found
Augmented sparse principal component analysis for high dimensional data
We study the problem of estimating the leading eigenvectors of a
high-dimensional population covariance matrix based on independent Gaussian
observations. We establish lower bounds on the rates of convergence of the
estimators of the leading eigenvectors under -sparsity constraints when an
loss function is used. We also propose an estimator of the leading
eigenvectors based on a coordinate selection scheme combined with PCA and show
that the proposed estimator achieves the optimal rate of convergence under a
sparsity regime. Moreover, we establish that under certain scenarios, the usual
PCA achieves the minimax convergence rate.Comment: This manuscript was written in 2007, and a version has been available
on the first author's website, but it is posted to arXiv now in its 2007
form. Revisions incorporating later work will be posted separatel
Sparse Representation of High Dimensional Data for Classification
In this thesis we propose the use of sparse Principal Component Analysis (PCA) for representing high dimensional data for classification. Sparse transformation reduces the data volume/dimensionality without loss of critical information, so that it can be processed efficiently and assimilated by a human. We obtained sparse representation of high dimensional dataset using Sparse Principal Component Analysis (SPCA) and Direct formulation of Sparse Principal Component Analysis (DSPCA). Later we performed classification using k Nearest Neighbor (kNN) Method and compared its result with regular PCA. The experiments were performed on hyperspectral data and various datasets obtained from University of California, Irvine (UCI) machine learning dataset repository. The results suggest that sparse data representation is desirable because sparse representation enhances interpretation. It also improves classification performance with certain number of features and in most of the cases classification performance is similar to regular PCA
Sparse principal component analysis for natural language processing
High dimensional data are rapidly growing in many different disciplines, particularly in natural language processing. The analysis of natural language processing requires working with high dimensional matrices of word embeddings obtained from text data. Those matrices are often sparse in the sense that they contain many zero elements. Sparse principal component analysis is an advanced mathematical tool for the analysis of high dimensional data. In this paper, we study and apply the sparse principal component analysis for natural language processing, which can effectively handle large sparse matrices. We study several formulations for sparse principal component analysis, together with algorithms for implementing those formulations. Our work is motivated and illustrated by a real text dataset. We find that the sparse principal component analysis performs as good as the ordinary principal component analysis in terms of accuracy and precision, while it shows two major advantages: faster calculations and easier interpretation of the principal components. These advantages are very helpful especially in big data situations
ECA: High Dimensional Elliptical Component Analysis in non-Gaussian Distributions
We present a robust alternative to principal component analysis (PCA) ---
called elliptical component analysis (ECA) --- for analyzing high dimensional,
elliptically distributed data. ECA estimates the eigenspace of the covariance
matrix of the elliptical data. To cope with heavy-tailed elliptical
distributions, a multivariate rank statistic is exploited. At the model-level,
we consider two settings: either that the leading eigenvectors of the
covariance matrix are non-sparse or that they are sparse. Methodologically, we
propose ECA procedures for both non-sparse and sparse settings. Theoretically,
we provide both non-asymptotic and asymptotic analyses quantifying the
theoretical performances of ECA. In the non-sparse setting, we show that ECA's
performance is highly related to the effective rank of the covariance matrix.
In the sparse setting, the results are twofold: (i) We show that the sparse ECA
estimator based on a combinatoric program attains the optimal rate of
convergence; (ii) Based on some recent developments in estimating sparse
leading eigenvectors, we show that a computationally efficient sparse ECA
estimator attains the optimal rate of convergence under a suboptimal scaling.Comment: to appear in JASA (T&M
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