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 of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of $p$-fusion 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.

References in corpus (7)

• The Convex Geometry of Linear Inverse Problems
• Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
• A Partial Derandomization of PhaseLift using Spherical Designs
• Minimization of the Probabilistic p-frame Potential
• Tight p-fusion frames
• Frame theory in directional statistics
• Phase Retrieval By Projections

Cited by in corpus (7)

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• Cubatures on Grassmannians: moments, dimension reduction, and related topics
• The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold
• Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices
• Preconditioning filter bank decompositions using structured normalized tight frames
• Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases
• Phase retrieval by random binary questions: Which complementary subspace is closer?