Signal reconstruction from the magnitude of subspace components

TL;DR

This paper introduces methods for signal reconstruction from subspace component norms, extending phase retrieval, with deterministic formulas, algorithms for erasure scenarios, and probabilistic guarantees for random subspace sets.

Contribution

It provides a closed-form reconstruction formula under cubature conditions, and a list-decoding algorithm for erasure scenarios, advancing phase retrieval techniques.

Findings

Deterministic reconstruction formula derived for cubature subspaces.

Algorithm for signal recovery with missing norm data using list decoding.

High-probability exact recovery with random subspaces of linear size via semidefinite programming.

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.