## Phase TransitionsHere we will show two phase transition diagrams for the Wirtinger Flow (WF) algorithm. ## Signal modelsWe consider two signal models: ## Random low-pass signals.Here, is given by with and and are i.i.d. . ## Random Gaussian signals.In this model, is a random complex Gaussian vector with i.i.d. entries of the form with and distributed as ; this can be expressed as where and are are i.i.d. so that the low-pass model is a ‘bandlimited’ version of this high-pass random model (variances are adjusted so that the expected power is the same). ## Measurement modelsWe perform simulations based on two different kinds of measurements: ## Gaussian measurements.We sample random complex Gaussian vectors and use measurements of the form . ## Coded diffraction patternsWe consider an acquisition model, where we collect data of the form In particular, we focus on a random model in which the 's are i.i.d. distributed, each having i.i.d. entries sampled from a distribution . An example of an admissible random variable is , where and are independent and distributed as We shall refer to this distribution as an ## Phase TransitionsBelow, we set , and generate one signal of each type which will be used in all the experiments. The empirical probability of succcess is an average over trials, where in each instance, we generate new random sampling vectors according to the Gaussian or CDP models. We declare a trial successful if the relative error of the reconstruction dist falls below . Figure below shows that around Gaussian phaseless measurements suffice for exact recovery with high probability via the Wirtinger flow algorithm. We also see that about six octanary patterns are sufficient. |