214 research outputs found
New bounds for circulant Johnson-Lindenstrauss embeddings
This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as
an important class of structured random JL embeddings, are formed by
randomizing the column signs of a circulant matrix generated by a random
vector. With the help of recent decoupling techniques and matrix-valued
Bernstein inequalities, we obtain a new bound
for Gaussian circulant JL embeddings.
Moreover, by using the Laplace transform technique (also called Bernstein's
trick), we extend the result to subgaussian case. The bounds in this paper
offer a small improvement over the current best bounds for Gaussian circulant
JL embeddings for certain parameter regimes and are derived using more direct
methods.Comment: 11 pages; accepted by Communications in Mathematical Science
Bregman Proximal Gradient Algorithm with Extrapolation for a class of Nonconvex Nonsmooth Minimization Problems
In this paper, we consider an accelerated method for solving nonconvex and
nonsmooth minimization problems. We propose a Bregman Proximal Gradient
algorithm with extrapolation(BPGe). This algorithm extends and accelerates the
Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive
global Lipschitz gradient continuity assumption needed in Proximal Gradient
algorithms (PG). The BPGe algorithm has higher generality than the recently
introduced Proximal Gradient algorithm with extrapolation(PGe), and besides,
due to the extrapolation step, BPGe converges faster than BPG algorithm.
Analyzing the convergence, we prove that any limit point of the sequence
generated by BPGe is a stationary point of the problem by choosing parameters
properly. Besides, assuming Kurdyka-{\'L}ojasiewicz property, we prove the
whole sequences generated by BPGe converges to a stationary point. Finally, to
illustrate the potential of the new method BPGe, we apply it to two important
practical problems that arise in many fundamental applications (and that not
satisfy global Lipschitz gradient continuity assumption): Poisson linear
inverse problems and quadratic inverse problems. In the tests the accelerated
BPGe algorithm shows faster convergence results, giving an interesting new
algorithm.Comment: Preprint submitted for publication, February 14, 201
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