17,114 research outputs found
Sparse PCA: Optimal rates and adaptive estimation
Principal component analysis (PCA) is one of the most commonly used
statistical procedures with a wide range of applications. This paper considers
both minimax and adaptive estimation of the principal subspace in the high
dimensional setting. Under mild technical conditions, we first establish the
optimal rates of convergence for estimating the principal subspace which are
sharp with respect to all the parameters, thus providing a complete
characterization of the difficulty of the estimation problem in term of the
convergence rate. The lower bound is obtained by calculating the local metric
entropy and an application of Fano's lemma. The rate optimal estimator is
constructed using aggregation, which, however, might not be computationally
feasible. We then introduce an adaptive procedure for estimating the principal
subspace which is fully data driven and can be computed efficiently. It is
shown that the estimator attains the optimal rates of convergence
simultaneously over a large collection of the parameter spaces. A key idea in
our construction is a reduction scheme which reduces the sparse PCA problem to
a high-dimensional multivariate regression problem. This method is potentially
also useful for other related problems.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1178 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Randomized Dimension Reduction on Massive Data
Scalability of statistical estimators is of increasing importance in modern
applications and dimension reduction is often used to extract relevant
information from data. A variety of popular dimension reduction approaches can
be framed as symmetric generalized eigendecomposition problems. In this paper
we outline how taking into account the low rank structure assumption implicit
in these dimension reduction approaches provides both computational and
statistical advantages. We adapt recent randomized low-rank approximation
algorithms to provide efficient solutions to three dimension reduction methods:
Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and
Localized Sliced Inverse Regression (LSIR). A key observation in this paper is
that randomization serves a dual role, improving both computational and
statistical performance. This point is highlighted in our experiments on real
and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized
eigendecompositon, low-rank, supervised, inverse regression, random
projections, randomized algorithms, Krylov subspace method
A new algorithm for estimating the effective dimension-reduction subspace
The statistical problem of estimating the effective dimension-reduction (EDR)
subspace in the multi-index regression model with deterministic design and
additive noise is considered. A new procedure for recovering the directions of
the EDR subspace is proposed. Under mild assumptions, -consistency of
the proposed procedure is proved (up to a logarithmic factor) in the case when
the structural dimension is not larger than 4. The empirical behavior of the
algorithm is studied through numerical simulations
Efficient Scalable Accurate Regression Queries in In-DBMS Analytics
Recent trends aim to incorporate advanced data analytics capabilities within DBMSs. Linear regression queries are fundamental to exploratory analytics and predictive modeling. However, computing their exact answers leaves a lot to be desired in terms of efficiency and scalability. We contribute a novel predictive analytics model and associated regression query processing algorithms, which are efficient, scalable and accurate. We focus on predicting the answers to two key query types that reveal dependencies between the values of different attributes: (i) mean-value queries and (ii) multivariate linear regression queries, both within specific data subspaces defined based on the values of other attributes. Our algorithms achieve many orders of magnitude improvement in query processing efficiency and nearperfect approximations of the underlying relationships among data attributes
AN ADAPTIVE COMPOSITE QUANTILE APPROACH TO DIMENSION REDUCTION
Sufficient dimension reduction [Li 1991] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an adaptive composite quantile approach. Compared to existing methods, (1) it requires minimal assumptions and is capable of revealing all dimension reduction directions; (2) it is robust against outliers and (3) it is structure-adaptive, thus more efficient. Asymptotic results are proved and numerical examples are provided, including a real data analysis
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