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This paper is concerned with stable phase retrieval for a family of phaseretrieval models we name "locally stable and conditionally connected" (LSCC)measurement schemes. For every signal $f$, we associate a correspondingweighted graph $G_f$, defined by the LSCC measurement scheme, and show that thephase retrievability of the signal $f$ is determined by the connectivity of$G_f$. We then characterize the phase retrieval stability of the signal $f$ bytwo measures that are commonly used in graph theory to quantify graphconnectivity: the Cheeger constant of $G_f$ for real valued signals, and thealgebraic connectivity of $G_f$ for complex valued signals.  We use our results to study the stability of two phase retrieval models thatcan be cast as LSCC measurement schemes, and focus on understanding for whichsignals the "curse of dimensionality" can be avoided. The first model wediscuss is a finite-dimensional model for locally supported measurements suchas the windowed Fourier transform. For signals "without large holes", we showthe stability constant exhibits only a mild polynomial growth in the dimension,in stark contrast with the exponential growth which uniform stability constantstend to suffer from; more precisely, in $R^d$ the constant grows proportionallyto $d^{1/2}$, while in $C^d$ it grows proportionally to $d$. We also show thegrowth of the constant in the complex case cannot be reduced, suggesting thatcomplex phase retrieval is substantially more difficult than real phaseretrieval. The second model we consider is an infinite-dimensional phaseretrieval problem in a principal shift invariant space. We show that despitethe infinite dimensionality of this model, signals with monotone exponentialdecay will have a finite stability constant. In contrast, the stability boundprovided by our results will be infinite if the signal's decay is polynomial.

Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements