5,152 research outputs found
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
Randomized hybrid linear modeling by local best-fit flats
The hybrid linear modeling problem is to identify a set of d-dimensional
affine sets in a D-dimensional Euclidean space. It arises, for example, in
object tracking and structure from motion. The hybrid linear model can be
considered as the second simplest (behind linear) manifold model of data. In
this paper we will present a very simple geometric method for hybrid linear
modeling based on selecting a set of local best fit flats that minimize a
global l1 error measure. The size of the local neighborhoods is determined
automatically by the Jones' l2 beta numbers; it is proven under certain
geometric conditions that good local neighborhoods exist and are found by our
method. We also demonstrate how to use this algorithm for fast determination of
the number of affine subspaces. We give extensive experimental evidence
demonstrating the state of the art accuracy and speed of the algorithm on
synthetic and real hybrid linear data.Comment: To appear in the proceedings of CVPR 201
A Nonconvex Projection Method for Robust PCA
Robust principal component analysis (RPCA) is a well-studied problem with the
goal of decomposing a matrix into the sum of low-rank and sparse components. In
this paper, we propose a nonconvex feasibility reformulation of RPCA problem
and apply an alternating projection method to solve it. To the best of our
knowledge, we are the first to propose a method that solves RPCA problem
without considering any objective function, convex relaxation, or surrogate
convex constraints. We demonstrate through extensive numerical experiments on a
variety of applications, including shadow removal, background estimation, face
detection, and galaxy evolution, that our approach matches and often
significantly outperforms current state-of-the-art in various ways.Comment: In the proceedings of Thirty-Third AAAI Conference on Artificial
Intelligence (AAAI-19
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