17,822 research outputs found
Robust Principal Component Analysis using Density Power Divergence
Principal component analysis (PCA) is a widely employed statistical tool used
primarily for dimensionality reduction. However, it is known to be adversely
affected by the presence of outlying observations in the sample, which is quite
common. Robust PCA methods using M-estimators have theoretical benefits, but
their robustness drop substantially for high dimensional data. On the other end
of the spectrum, robust PCA algorithms solving principal component pursuit or
similar optimization problems have high breakdown, but lack theoretical
richness and demand high computational power compared to the M-estimators. We
introduce a novel robust PCA estimator based on the minimum density power
divergence estimator. This combines the theoretical strength of the
M-estimators and the minimum divergence estimators with a high breakdown
guarantee regardless of data dimension. We present a computationally efficient
algorithm for this estimate. Our theoretical findings are supported by
extensive simulations and comparisons with existing robust PCA methods. We also
showcase the proposed algorithm's applicability on two benchmark datasets and a
credit card transactions dataset for fraud detection
Decentralization Estimators for Instrumental Variable Quantile Regression Models
The instrumental variable quantile regression (IVQR) model (Chernozhukov and
Hansen, 2005) is a popular tool for estimating causal quantile effects with
endogenous covariates. However, estimation is complicated by the non-smoothness
and non-convexity of the IVQR GMM objective function. This paper shows that the
IVQR estimation problem can be decomposed into a set of conventional quantile
regression sub-problems which are convex and can be solved efficiently. This
reformulation leads to new identification results and to fast, easy to
implement, and tuning-free estimators that do not require the availability of
high-level "black box" optimization routines
Algorithms and Hardness for Robust Subspace Recovery
We consider a fundamental problem in unsupervised learning called
\emph{subspace recovery}: given a collection of points in ,
if many but not necessarily all of these points are contained in a
-dimensional subspace can we find it? The points contained in are
called {\em inliers} and the remaining points are {\em outliers}. This problem
has received considerable attention in computer science and in statistics. Yet
efficient algorithms from computer science are not robust to {\em adversarial}
outliers, and the estimators from robust statistics are hard to compute in high
dimensions.
Are there algorithms for subspace recovery that are both robust to outliers
and efficient? We give an algorithm that finds when it contains more than a
fraction of the points. Hence, for say this estimator
is both easy to compute and well-behaved when there are a constant fraction of
outliers. We prove that it is Small Set Expansion hard to find when the
fraction of errors is any larger, thus giving evidence that our estimator is an
{\em optimal} compromise between efficiency and robustness.
As it turns out, this basic problem has a surprising number of connections to
other areas including small set expansion, matroid theory and functional
analysis that we make use of here.Comment: Appeared in Proceedings of COLT 201
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