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...essing has grown into a active field of research and in particular proximal splitting methods [6], [7], [8] have been used to great effect, e.g. in audio inpainting [2], [1] and sparse representation =-=[12]-=-. In those cases, optimization techniques are applied directly to the signal or its time-frequency representation. In this contribution, we apply optimization techniques to shape the building blocks o...

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...lowed locations of the nonzero entries of the analyzed signal, which are more heavily restricted in the group sparse case. C. Group sparse regularization penalties A standard approach to promote group sparsity is to penalize the mixed `2-`1 norm of the signal restricted to the groups G: Φ1(y) = ∑ g∈G ‖yg‖2, (3) where yg ∈ Rd denotes the vector whose entries are the same as y ∈ Rd on the index set g ∈ G, and zeros elsewhere. This penalty is also known as the group lasso, and its ability to recover group sparse signals been studied extensively [14], including the case of overlapping groups [15]–[17]. To better approximate the ideal `0 measure of sparsity, we also consider the following non-convex variant: Φp(y) = 1 p ∑ g∈G ‖yg‖p2, 0 < p ≤ 1. (4) In our setting, we may give a more compact representation of the penalties in (3) and (4) by introducing an expansion operator E : Rd → R(d−n+1)×n defined by [ y1 y2 · · · yd ]T 7→ 1√ n y1 y2 · · · yd−n+1... ... ... yn yn+1 · · · yd T , where the rows of Ey collect the coefficients of y belonging to each group of n consecutive indices. Up to a scaling, we may recast (3) and (4) as Φp(y) = ‖Ey‖p2,p for 0 < p ≤ 1, where ‖U‖p2,p := 1p ∑ i ‖ui...

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... be prevented or reduced by taking into account the behavior of the neighboring time-frequency atoms [14, 8] or working in time-frequency blocks of atoms, rather than individual atoms [26] – see also =-=[18, 21, 3, 19, 22, 10]-=- for recent E-mail : ibayram@itu.edu.tr work in this context. The reason for this is that the time-frequency distribution of audio follows predictable patterns with well-formed groups. This is especia...

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...able over treating the atoms as independent entries. This work is supported by Academy of Finland (project # 256961 / 284671). Group sparsity functions have been proposed for general sparse models in =-=[8, 9, 10, 11, 12]-=-. Recent studies on e.g. group LASSO regularisation also permit overlapping and variable-size groups. Models for NMF in particular were demonstrated in [13] and [14] for modelling speech with multiple...

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...rocessing has grown into a active field of research and in particular proximal splitting methods [13, 12, 14] have been used to great effect, e.g. in audio inpainting [2, 1] and sparse representation =-=[25]-=-. In those cases, optimization techniques are applied directly to the signal or its time-frequency representation. In this contribution, we apply optimization techniques to shape the building blocks o...

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... the combination of total variation (TV) via the Fast Composite Splitting Algorithms (FCSA) scheme [51]. Note that other algorithms may be used to solve the forest sparsity problems, e.g. [32], [44], =-=[52]-=-, but determining the optimal algorithm for forest sparsity is not the scope of this article. FISTA [12] is a accelerated version of proximal method which minimizes the object function with the follow...

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...f the union of two Gabor frames, each adapted to the “morphological layer”, for the signal time-frequency representation in the problem of source separation. Secondly, we link the WindowedGroup-Lasso =-=[7]-=- to the problem of source separation to obtain a more reliable sparse representation. Windowed group-Lasso is a convenient way to take into account some neighborhood information for a structured spars...