It is known in the standard l2 distance that the closest Parseval frame to a given frame {Φi}mi=1 ⊂ ℜ d is {S-1/2(Φi)}mi=1 where S is the frame operator of the original frame. We show this is also the case for probabilistic frames in the 2-Wasserstein distance. In the process we develop some regularity properties for the map taking a probabilistic frame to its closest Parseval frame.
A finite frame is said to be scalable if its vectors can be rescaled so that the resulting set of vectors is a tight frame. The theory of scalable frame has been extended to the setting of Laplacian pyramids which are based on (rectangular) paraunitary matrices whose column vectors are Laurent polynomial vectors. This is equivalent to scaling the polyphase matrices of the associated filter banks. Consequently, tight wavelet frames can be constructed by appropriately scaling the columns of these paraunitary matrices by diagonal matrices whose diagonal entries are square magnitude of Laurent polynomials. In this paper we present examples of tight wavelet frames constructed in this manner and discuss some of their properties in comparison to the (non tight) wavelet frames they arise from.
KEYWORDS: Radon, Mathematics, Matrices, Condition numbers, Vector spaces, Signal processing, Linear algebra, Wavelets, Current controlled current source
The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight frame. In this paper we define m−scalability, a refinement of scalability based on the number of non-zero weights used in the rescaling process. We enlighten a close connection between this notion and elements from convex geometry. Another focus lies in the topology of scalable frames. In particular, we prove that the set of scalable frames with “usual” redundancy is nowhere dense in the set of frames.
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