HOSVDHigher-Order Singular Value Decomposition
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The HOSVD of the received signal (12) can be written as
The HOSVD is then exploited to evaluate the signal subspace of the tensor-based received signal matrix.
The general problem of grouping ICs coming from HOSVD followed by ICA benefits from a special decomposition of [J.sub.Ed]([z.sub.ng]) into positive terms, hereafter called interactivities (ITIs).
The above source separation method depends crucially on the approximation of the expectation of any function /(z) of z-function: E[/(z)] (e.g., the cumulants (2), the HOSVD matrix M(z) (4), the negentropy (5), and its interactivities (8)), through its estimation obtained by sample averages: [bar.f(z)].
Estimation of the Gaussian Subspace by HOSVD. Following a rejection of [H.sub.G], the next step is to estimate the Gaussian subspace [W.sub.g] and its dimension [N.sub.g].
where [[psi].sub.HOSVD] is the HOSVD base estimator.
Let us denote {[P.sub.n]} as the stack Z [member of] [L.sup.nxnxK]; the HOSVD of the stack can then be defined as [20]
After applying the HOSVD transform, the patches can be estimated by nullifying the coefficients under the assumption that the coefficients of the clean image have a sparse distribution.
The importance of optimizing only over the positive and negative observations is demonstrated by the performance of the HOSVD method, which is significantly poorer.
Similarly to 2D arrays, subspace methods for three-mode arrays are based on a rank approximation of the HOSVD. Consider three-mode array (6) and its decomposition into two three-mode arrays
[N.sub.n] can be expressed in terms of an "economy size" HOSVD in the following way [11,12]:
After looking at several combinations of three-mode ranks for the HOSVD subspace method, we have chosen one with rank(2, 2, 2).