Signal Reconstruction From the Magnitude of Subspace Components

Abstract

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.

Cubatures on Grassmannians: Moments, Dimension Reduction, and Related Topics

“…cf. [14] and also [4]. Since the constant functions are contained in Hom t (G k,d ), any cubature must satisfy n j=1 ω j = 1.…”

“…Theorem 3 ( [14,4]). If n j=1 ω j = 1 and (14) holds with equality, then {(P j , ω j )} n j=1 is a cubature for Hom t (G k,d ).…”

“…To proceed let us denote n := dim(Pol t (G k,d ∪ G d−k,d )). According to [28], see also [4], there are {X j } n j=1 ⊂ G k,d ∪ G d−k,d and {c j } n j=1 ⊂ R such that f (P ) = n j=1 c j trace(X j P ) t | G k,d ∪ G d−k,d , P ∈ G k,d ∪ G d−k,d .…”

“…Intuitively, good samplings for function approximation should well cover the underlying space, and, in Section 5. 4, we recapitulate that sequences of low cardinality Grassmannian cubatures are asymptotically optimal coverings, cf. [13].…”

“…Therefore, we then discuss in Section 5.1 numerical constructions of cubatures in Grassmannians by minimizing the worst case integration error of polynomials, cf. [14,4]. In Section 5.2, we go beyond polynomials and briefly discuss sequences of low cardinality cubatures that yield optimal worst case integration error rates for Bessel potential functions, cf.…”