Splitted Wavelet Differential Inclusion
Primary Area: metric learning, kernel learning, and sparse coding
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Keywords: Wavelet, Differential Inclusion, Weak Signal, EEG
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TL;DR: We propose a Wavelet Differential Inclusion method to achieve more accurate estimation of the signal and apply them to neuroimaing analysis.
Abstract: Wavelet Shrinkage typically selects only a small proportion of large coefficients via soft or hard thresholding, since the \emph{strong signal} composed by these coefficients has more semantic meaning than others. Typical examples include the object's shape in the image or the burst activity in the low $\beta$ band in Parkinson's Disease. However, it has been found that there also exists \emph{weak signal} that should not be ignored. Such a weak signal refers to the set of small coefficients, which in the above examples \emph{resp.} correspond to the texture of an image or the non-burst/tonic activity in Parkinson's Disease. Although it is not as interpretable as the strong signal, ignorance of it may miss information in signal reconstruction. Existing methods either suffered from failing to disentangle the strong signal apart with a too small threshold parameter, or inaccurate estimation of the whole signal (\emph{i.e.}, strong and weak signals) due to the bias/errors in the strong signal and over-smoothing of the weak signal. To resolve these problems, we propose a \emph{Splitted Wavelet Differential Inclusion}, which is provable to achieve better estimation on both the strong signal and the whole signal than Wavelet Shrinkage. Specifically, equipped with an $\ell_2$ splitting mechanism, we obtain the solution path from the differential inclusion of a couple of parameters, of which the sparse one can remove bias in estimating the strong signal and the dense parameter can additionally capture the weak signal with the $\ell_2$ shrinkage. The utility of our method is demonstrated by the improved accuracy in a numerical experiment and moreover the additional findings of tonic activity in Parkinson's Disease.
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Submission Number: 4663
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