307 research outputs found
Blind Source Separation with Compressively Sensed Linear Mixtures
This work studies the problem of simultaneously separating and reconstructing
signals from compressively sensed linear mixtures. We assume that all source
signals share a common sparse representation basis. The approach combines
classical Compressive Sensing (CS) theory with a linear mixing model. It allows
the mixtures to be sampled independently of each other. If samples are acquired
in the time domain, this means that the sensors need not be synchronized. Since
Blind Source Separation (BSS) from a linear mixture is only possible up to
permutation and scaling, factoring out these ambiguities leads to a
minimization problem on the so-called oblique manifold. We develop a geometric
conjugate subgradient method that scales to large systems for solving the
problem. Numerical results demonstrate the promising performance of the
proposed algorithm compared to several state of the art methods.Comment: 9 pages, 2 figure
Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space
In this paper, we introduce the notion of generalized -stationarity
for a class of nonconvex and nonsmooth composite minimization problems on
compact Riemannian submanifold embedded in Euclidean space. To find a
generalized -stationarity point, we develop a family of Riemannian
gradient-type methods based on the Moreau envelope technique with a decreasing
sequence of smoothing parameters, namely Riemannian smoothing gradient and
Riemannian smoothing stochastic gradient methods. We prove that the Riemannian
smoothing gradient method has the iteration complexity of
for driving a generalized -stationary
point. To our knowledge, this is the best-known iteration complexity result for
the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian
smoothing stochastic gradient method, one can achieve the iteration complexity
of for driving a generalized -stationary
point. Numerical experiments are conducted to validate the superiority of our
algorithms
Smoothing algorithms for nonsmooth and nonconvex minimization over the stiefel manifold
We consider a class of nonsmooth and nonconvex optimization problems over the
Stiefel manifold where the objective function is the summation of a nonconvex
smooth function and a nonsmooth Lipschitz continuous convex function composed
with an linear mapping. We propose three numerical algorithms for solving this
problem, by combining smoothing methods and some existing algorithms for smooth
optimization over the Stiefel manifold. In particular, we approximate the
aforementioned nonsmooth convex function by its Moreau envelope in our
smoothing methods, and prove that the Moreau envelope has many favorable
properties. Thanks to this and the scheme for updating the smoothing parameter,
we show that any accumulation point of the solution sequence generated by the
proposed algorithms is a stationary point of the original optimization problem.
Numerical experiments on building graph Fourier basis are conducted to
demonstrate the efficiency of the proposed algorithms.Comment: 22 page
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