Matrix recovery is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means to recover almost all the @math from @math where @math . We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in @math ? For the case where both @math and @math are algebraic varieties, we use the tools from algebraic geometry to study the question and present some results to address it under many different settings.
Abstract 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 4 M − 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. We will construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using phase or its estimation. This verifies a longstanding conjecture of the speech processing community. Consider a scenario in which an unknown signal is transformed by a known linear operator, and then the pointwise absolute value of the unknown output function is reported. This scenario appears in several applications, and the goal is to recover the unknown signal – this is called phase retrieval. Phase retrieval has been a popular subject of research in the last few years, both in determining whether complete information is available with a given linear operator and in finding efficient and stable phase retrieval algorithms in the cases where complete information is available. Interestingly, there are a few ways to measure information completeness, and each way appears to be governed by a phase transition of sorts. This chapter will survey the state of the art with some of these phase transitions, and identify a few open problems for further research. I construct a positive-operator-valued measure (POVM) which has 2d rank-1 elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph 0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in d dimensions, it must have at least 3d-2 elements. I also show that, in a POVM which is informationally complete for all pure states in d dimensions, for any vector there must be at least 2d-1 POVM elements which do not annihilate that vector.