918 research outputs found
The Convergence Guarantees of a Non-convex Approach for Sparse Recovery
In the area of sparse recovery, numerous researches hint that non-convex
penalties might induce better sparsity than convex ones, but up until now those
corresponding non-convex algorithms lack convergence guarantees from the
initial solution to the global optimum. This paper aims to provide performance
guarantees of a non-convex approach for sparse recovery. Specifically, the
concept of weak convexity is incorporated into a class of sparsity-inducing
penalties to characterize the non-convexity. Borrowing the idea of the
projected subgradient method, an algorithm is proposed to solve the non-convex
optimization problem. In addition, a uniform approximate projection is adopted
in the projection step to make this algorithm computationally tractable for
large scale problems. The convergence analysis is provided in the noisy
scenario. It is shown that if the non-convexity of the penalty is below a
threshold (which is in inverse proportion to the distance between the initial
solution and the sparse signal), the recovered solution has recovery error
linear in both the step size and the noise term. Numerical simulations are
implemented to test the performance of the proposed approach and verify the
theoretical analysis.Comment: 33 pages, 7 figure
Robust computation of linear models by convex relaxation
Consider a dataset of vector-valued observations that consists of noisy
inliers, which are explained well by a low-dimensional subspace, along with
some number of outliers. This work describes a convex optimization problem,
called REAPER, that can reliably fit a low-dimensional model to this type of
data. This approach parameterizes linear subspaces using orthogonal projectors,
and it uses a relaxation of the set of orthogonal projectors to reach the
convex formulation. The paper provides an efficient algorithm for solving the
REAPER problem, and it documents numerical experiments which confirm that
REAPER can dependably find linear structure in synthetic and natural data. In
addition, when the inliers lie near a low-dimensional subspace, there is a
rigorous theory that describes when REAPER can approximate this subspace.Comment: Formerly titled "Robust computation of linear models, or How to find
a needle in a haystack
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