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Tight pfusion frames
"... Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann s ..."
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Cited by 8 (4 self)
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Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the pfusion frame potential, derive bounds for its value, and discuss the connections to tight pfusion frames.
Optimally Sparse Frames
 IEEE Trans. Inform. Theory
, 2011
"... Dedicated to the memory of Nigel J. Kalton, who was a great person, friend, and mathematician. ..."
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Dedicated to the memory of Nigel J. Kalton, who was a great person, friend, and mathematician.
Nonorthogonal fusion frames and the sparsity of fusion frame operators
 J. Fourier Anal. Appl
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Prime tight frames
 Adv. Comput. Math
"... Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constr ..."
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Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constructed from the spectral tetris method. As a byproduct of this last result, we obtain a characterization of when the spectral tetris construction works for redundancies below two.
Necessary and sufficient conditions to perform Spectral Tetris, Linear Algebra Appl
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Sparsity and spectral properties of dual frames
"... Abstract: We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than n 2 nonvanishing entries, where n denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expre ..."
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Abstract: We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than n 2 nonvanishing entries, where n denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight.
Manifold Matching: Joint Optimization of Fidelity and Commensurability
, 2010
"... Fusion and inference from multiple and massive disparate data sources – the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies – has many and varied aspects which are currently the target of intense research an ..."
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Fusion and inference from multiple and massive disparate data sources – the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies – has many and varied aspects which are currently the target of intense research and development. One aspect of the overall challenge is manifold matching – identifying embeddings of multiple disparate data spaces into the same lowdimensional space where joint inference can be pursued. We investigate this manifold matching task from the perspective of jointly optimizing the fidelity of the embeddings and their commensurability with one another, with a specific statistical inference exploitation task in mind. Our results demonstrate when and why our joint optimization methodology is superior to either version of separate optimization. The methodology is illustrated with simulations and an application in document matching.
FUSION FRAMES AND THE RESTRICTED ISOMETRY PROPERTY
"... Abstract. We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equiisoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restric ..."
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Abstract. We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equiisoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restricted isometry property. 1.