  Analysis of high dimensional noisy data is of essence across a variety of
research fields. Feature selection techniques are designed to find the relevant
feature subset that can facilitate classification or pattern detection.
Traditional (supervised) feature selection methods utilize label information to
guide the identification of relevant feature subsets. In this paper, however,
we consider the unsupervised feature selection problem. Without the label
information, it is particularly difficult to identify a small set of relevant
features due to the noisy nature of real-world data which corrupts the
intrinsic structure of the data. Our Gradient-based Laplacian Feature Selection
(GLFS) selects important features by minimizing the variance of the Laplacian
regularized least squares regression model. With $\ell_1$ relaxation, GLFS can
find a sparse subset of features that is relevant to the Laplacian manifolds.
Extensive experiments on simulated, three real-world object recognition and two
computational biology datasets, have illustrated the power and superior
performance of our approach over multiple state-of-the-art unsupervised feature
selection methods. Additionally, we show that GLFS selects a sparser set of
more relevant features in a supervised setting outperforming the popular
elastic net methodology.
