729 research outputs found
On Convex Envelopes and Regularization of Non-Convex Functionals without moving Global Minima
We provide theory for the computation of convex envelopes of non-convex
functionals including an l2-term, and use these to suggest a method for
regularizing a more general set of problems. The applications are particularly
aimed at compressed sensing and low rank recovery problems but the theory
relies on results which potentially could be useful also for other types of
non-convex problems. For optimization problems where the l2-term contains a
singular matrix we prove that the regularizations never move the global minima.
This result in turn relies on a theorem concerning the structure of convex
envelopes which is interesting in its own right. It says that at any point
where the convex envelope does not touch the non-convex functional we
necessarily have a direction in which the convex envelope is affine.Comment: arXiv admin note: text overlap with arXiv:1609.0937
ESPRIT for multidimensional general grids
We present a new method for complex frequency estimation in several
variables, extending the classical (1d) ESPRIT-algorithm. We also consider how
to work with data sampled on non-standard domains (i.e going beyond
multi-rectangles)
A Non-Convex Relaxation for Fixed-Rank Approximation
This paper considers the problem of finding a low rank matrix from
observations of linear combinations of its elements. It is well known that if
the problem fulfills a restricted isometry property (RIP), convex relaxations
using the nuclear norm typically work well and come with theoretical
performance guarantees. On the other hand these formulations suffer from a
shrinking bias that can severely degrade the solution in the presence of noise.
In this theoretical paper we study an alternative non-convex relaxation that
in contrast to the nuclear norm does not penalize the leading singular values
and thereby avoids this bias. We show that despite its non-convexity the
proposed formulation will in many cases have a single local minimizer if a RIP
holds. Our numerical tests show that our approach typically converges to a
better solution than nuclear norm based alternatives even in cases when the RIP
does not hold
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