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SIGNAL RECONSTRUCTION FROM THE MAGNITUDE OF SUBSPACE COMPONENTS
"... Abstract. We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number ..."
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Abstract. We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of pfusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program. 1.
CUBATURES AND DESIGNS IN UNIONS OF GRASSMANNIANS
"... Abstract. The Grassmannian can be considered as the set of orthogonal projectors of fixed rank in the ddimensional Euclidean space. Cubatures and designs on the Grassmannian have been wellstudied in the recent literature. On the other hand, particular sets of projectors with potentially varying ..."
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Abstract. The Grassmannian can be considered as the set of orthogonal projectors of fixed rank in the ddimensional Euclidean space. Cubatures and designs on the Grassmannian have been wellstudied in the recent literature. On the other hand, particular sets of projectors with potentially varying ranks have been used in signal processing under the name fusion frames. The relations between cubatures, designs, and fusion frames have already been investigated in the literature when the rank was held fixed. Here, we introduce cubatures and designs in unions of Grassmannians and discuss the relations towards fusion frames with varying ranks. We characterize cubatures and designs in unions of Grassmannians by means of the fusion frame potential matching a certain lower bound, and we present parametric families of symmetric designs in unions of Grassmannians. 1.
ON TIGHT GENERALIZED FRAMES
"... Abstract. Frames can be thought of as collections of rankone, positive semidefinite operators that, if tight, enable signal decompositions like orthogonal bases. One refers to generalized frames if the rankone constraint is withdrawn. Given a generalized frame, we construct in a linear fashion ..."
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Abstract. Frames can be thought of as collections of rankone, positive semidefinite operators that, if tight, enable signal decompositions like orthogonal bases. One refers to generalized frames if the rankone constraint is withdrawn. Given a generalized frame, we construct in a linear fashion a tight generalized frame whose elements have equal norm and are close to the original frame. For integers p, we also introduce stochastic generalized pframes and verify that common random matrices used in compressed sensing satisfy tightness conditions. We then suggest a refinement of the notion of frame redundancy and discuss a few supporting examples. 1.
With r(d
"... Abstract. Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. ..."
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Abstract. Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed.
Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables
"... Abstract. Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With r(d, n): = ( n+d−1 n, the firs ..."
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Abstract. Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With r(d, n): = ( n+d−1 n, the first method generates, for each m ≥ n, two families of tight frames in R r(d,n) with r(d + 1, m) elements. The second method generates a tight frame in R r(d,N) with 1+N ×r(d+1, N) vectors. All frame elements are given in explicit formulas.