Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2, 3. Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer-Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.

In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient phase retrieval algorithms from such measurements. For the case of complex M -dimensional signals, we construct a measurement ensemble of size 4M − 4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of "almost" injective intensity measurements, and we characterize such ensembles. Later, we show that phase retrieval from M + 1 almost injective intensity measurements is NP-hard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.

Plant phenology will likely shift with climate change, but how temperature and/or moisture regimes will control phenological responses is not well understood. This is particularly true in Mediterranean climate ecosystems where the warmest temperatures and greatest moisture availability are seasonally asynchronous. We examined plant phenological responses at both the population and community levels to four climate treatments (control, warming, drought, and warming plus additional precipitation) embedded within three prairies across a 520 km latitudinal Mediterranean climate gradient within the Pacific Northwest, USA. At the population level, we monitored flowering and abundances in spring 2017 of eight range‐restricted focal species planted both within and north of their current ranges. At the community level, we used normalized difference vegetation index (NDVI) measured from fall 2016 to summer 2018 to estimate peak live biomass, senescence, seasonal patterns, and growing season length. We found that warming exerted a stronger control than our moisture manipulations on phenology at both the population and community levels. Warming advanced flowering regardless of whether a species was within or beyond its current range. Importantly, many of our focal species had low abundances, particularly in the south, suggesting that establishment, in addition to phenological shifts, may be a strong constraint on their future viability. At the community level, warming advanced the date of peak biomass regardless of site or year. The date of senescence advanced regardless of year for the southern and central sites but only in 2018 for the northern site. Growing season length contracted due to warming at the southern and central sites (~3 weeks) but was unaffected at the northern site. Our results emphasize that future temperature changes may exert strong influence on the timing of a variety of plant phenological events, especially those events that occur when temperature is most limiting, even in seasonally water‐limited Mediterranean ecosystems.

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