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56
Phase Retrieval via Matrix Completion
, 2011
"... This paper develops a novel framework for phase retrieval, a problem which arises in Xray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to ..."
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Cited by 71 (10 self)
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This paper develops a novel framework for phase retrieval, a problem which arises in Xray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complexvalued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noiseaware algorithms are stable in the sense that the reconstruction degrades gracefully as the signaltonoise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.
Testing the limits of quantum mechanics: motivation, state of play, prospects
, 2002
"... I present the motivation for experiments which attempt to generate, and verify the existence of, quantum superpositions of two or more states which are by some reasonable criterion ‘macroscopically’ distinct, and show that various a priori objections to this programme made in the literature are flaw ..."
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Cited by 52 (0 self)
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I present the motivation for experiments which attempt to generate, and verify the existence of, quantum superpositions of two or more states which are by some reasonable criterion ‘macroscopically’ distinct, and show that various a priori objections to this programme made in the literature are flawed. I review the extent to which such experiments currently exist in the areas of freespace molecular diffraction, magnetic biomolecules, quantum optics and Josephson devices, and sketch possible future lines of development of the programme.
Phase Retrieval from Coded Diffraction Patterns
, 2013
"... This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the inten ..."
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Cited by 21 (5 self)
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This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
, 2014
"... We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This pape ..."
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Cited by 18 (4 self)
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We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a nearlinear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.
On Counterfactuals and Contextuality
 in AIP Conference Proceedings 750. Foundations of Probability and Physics3, edited by A. Khrennikov, American Institute of Physics
, 2005
"... Abstract. Counterfactual reasoning and contextuality is defined and critically evaluated with regard to its nonempirical content. To this end, a uniqueness property of states, explosion views and link observables are introduced. If only a single context associated with a particular maximum set of ob ..."
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Cited by 8 (4 self)
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Abstract. Counterfactual reasoning and contextuality is defined and critically evaluated with regard to its nonempirical content. To this end, a uniqueness property of states, explosion views and link observables are introduced. If only a single context associated with a particular maximum set of observables can be operationalized, then a context translation principle resolves measurements of different contexts. COUNTERFACTUALS With the rise of quantum mechanics [1, 2, 3, 4] physics proper entered an ancient and sometimes fierce debate in theology and philosophy: the controversy between realism versus idealism. Whereas realism has been subsumed by the proposition that [5] “some entities sometimes exist without being experienced by any finite mind, ” idealism put forward that “we have not the faintest reason for believing in the existence of unexperienced entities. [[Realism]] has been adopted... solely because it simplifies our view of the universe. ” And whereas these issues can be considered nonoperational and thus metaphysical or even ideological, it is also true that they have inspired a great number of minds, to the effect of stimulating new approaches to quantum mechanics, revealing many theoretical details, quantum phenomena and quantum technologies. The KochenSpecker theorem [6], for example, was motivated from the onset by scholasticism, as in an early programmatic article [7] Ernst Specker related the discussion on the foundations of quantum mechanics to scholastic
The Content and Acquisition of Lexical Concepts
, 2006
"... This thesis aims to develop a psychologically plausible account of concepts by integrating key insights from philosophy (on the metaphysical basis for concept possession) and psychology (on the mechanisms underlying concept acquisition). I adopt an approach known as informational atomism, develope ..."
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Cited by 6 (0 self)
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This thesis aims to develop a psychologically plausible account of concepts by integrating key insights from philosophy (on the metaphysical basis for concept possession) and psychology (on the mechanisms underlying concept acquisition). I adopt an approach known as informational atomism, developed by Jerry Fodor. Informational atomism is the conjunction of two theses: (i) informational semantics, according to which conceptual content is constituted exhaustively by nomological mind–world relations; and (ii) conceptual atomism, according to which (lexical) concepts have no internal structure. I argue that informational semantics needs to be supplemented by allowing contentconstitutive rules of inference (“meaning postulates”). This is because the content of one important class of concepts, the logical terms, is not plausibly informational. And since, it is argued, no principled distinction can be drawn between logical concepts and the rest, the problem that this raises is a general one.
Criteria for continuousvariable quantum teleportation”, eprint quantph/9910030
, 1999
"... We derive an experimentally testable criterion for the teleportation of quantum states of continuous variables. This criterion is especially relevant to the recent experiment of Furusawa et al. [Science 282, 706 (1998)] where an inputoutput fidelity of 0.58 ± 0.02 was achieved for optical coherent ..."
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Cited by 5 (1 self)
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We derive an experimentally testable criterion for the teleportation of quantum states of continuous variables. This criterion is especially relevant to the recent experiment of Furusawa et al. [Science 282, 706 (1998)] where an inputoutput fidelity of 0.58 ± 0.02 was achieved for optical coherent states. Our derivation demonstrates that fidelities greater than 1/2 could not have been achieved through the use of a classical channel alone; quantum entanglement was a crucial ingredient in the experiment. 1
Is it true; or is it false; or somewhere in between? The logic of quantum theory. Contempory Phys
, 2005
"... The paper contains a relatively nontechnical summary of some recent work by the author and Jeremy Butterfield. The goal is to find a way of assigning meaningful truth values to propositions in quantum theory: something that is not possible in the normal, instrumentalist interpretation. The key math ..."
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Cited by 4 (3 self)
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The paper contains a relatively nontechnical summary of some recent work by the author and Jeremy Butterfield. The goal is to find a way of assigning meaningful truth values to propositions in quantum theory: something that is not possible in the normal, instrumentalist interpretation. The key mathematical tool is presheaf theory where, multivalued, contextual truth values arise naturally. We show how this can be applied to quantum theory, with the ‘contexts ’ chosen to be Boolean subalgebras of the set of all projection operators.