5,803 research outputs found
Multiscale Decompositions and Optimization
In this paper, the following type Tikhonov regularization problem will be
systematically studied: [(u_t,v_t):=\argmin_{u+v=f} {|v|_X+t|u|_Y},] where
is a smooth space such as a \BV space or a Sobolev space and is the pace
in which we measure distortion. Examples of the above problem occur in
denoising in image processing, in numerically treating inverse problems, and in
the sparse recovery problem of compressed sensing. It is also at the heart of
interpolation of linear operators by the real method of interpolation. We shall
characterize of the minimizing pair for
(X,Y)=(L_2(\Omega),\BV(\Omega)) as a primary example and generalize Yves
Meyer's result in [11] and Antonin Chambolle's result in [6]. After that, the
following multiscale decomposition scheme will be studied:
[u_{k+1}:=\argmin_{u\in \BV(\Omega)\cap L_2(\Omega)}
{1/2|f-u|^2_{L_2}+t_{k}|u-u_k|_{\BV}},] where and is a bounded
Lipschitz domain in . This method was introduced by Eitan Tadmor et al.
and we will improve the convergence result in \cite{Tadmor}. Other pairs
such as and will also be
mentioned. In the end, the numerical implementation for
(X,Y)=(L_2(\Omega),\BV(\Omega)) and the corresponding convergence results
will be given.Comment: 33 page
A scale-based approach to finding effective dimensionality in manifold learning
The discovering of low-dimensional manifolds in high-dimensional data is one
of the main goals in manifold learning. We propose a new approach to identify
the effective dimension (intrinsic dimension) of low-dimensional manifolds. The
scale space viewpoint is the key to our approach enabling us to meet the
challenge of noisy data. Our approach finds the effective dimensionality of the
data over all scale without any prior knowledge. It has better performance
compared with other methods especially in the presence of relatively large
noise and is computationally efficient.Comment: Published in at http://dx.doi.org/10.1214/07-EJS137 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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