Cheeger Bounds for Stable Phase Retrieval in Reproducing Kernel Hilbert Spaces

TL;DR

This paper introduces a kernel Cheeger constant to analyze the stability of phase retrieval in reproducing kernel Hilbert spaces, providing unified bounds and characterizations applicable to various measurement domains and transforms.

Contribution

It proposes a novel kernel Cheeger constant for stability analysis, unifies finite and infinite-dimensional cases, and offers new criteria for wavelet phase retrieval.

Findings

Established lower and upper bounds for stability using the kernel Cheeger constant.

Bound the Cheeger constant in wavelet phase retrieval via data-dependent graph Cheeger constants.

Derived a sufficient criterion for wavelet sign retrieval in arbitrary dimensions.

Abstract

Phase retrieval seeks to reconstruct a signal from phaseless intensity measurements and, in applications where measurements contain errors, demands stable reconstruction. We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Motivated by Grohs-Rathmair's Cheeger-type estimate for Gabor phase retrieval, we introduce a kernel Cheeger constant that quantifies connectedness relative to kernel localization. This notion yields a clean stability certificate: we establish a unified lower bound over both real and complex fields, and in the real case also an upper bound, each in terms of the reciprocal kernel Cheeger constant. Our framework treats finite- and infinite-dimensional settings uniformly and covers discrete, semi-discrete, and continuous measurement domains. For generalized wavelet phase retrieval from (semi-)discrete frames, we bound the kernel Cheeger constant by the Cheeger constant of a data-dependent weighted graph. We further characterize phase retrievability for generalized wavelet transforms and derive a simple sufficient criterion for wavelet sign retrieval in arbitrary dimension for transforms associated with irreducibly admissible matrix groups.