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Open Access Publications from the University of California

Scholarly Works (30 results)

  • Article
  • Peer Reviewed
Faculty (2011)
Random illumination is proposed to enforce absolute uniqueness and resolve all types of ambiguity, trivial or nontrivial, from phase retrieval. Almost sure irreducibility is proved for any complex-valued object of a full rank support. While the new irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a novel perspective and an effective method for phase retrieval. In particular, almost sure uniqueness, up to a global phase, is proved for complex-valued objects under general two-point conditions. Under a tight sector constraint absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases. Under a magnitude constraint with random amplitude illumination, uniqueness modulo global phase is proved to hold with probability exponentially close to unity as object sparsity increases. For general complex-valued objects without any constraint, almost sure uniqueness up to global phase is established with two sets of Fourier magnitude data under two independent illuminations. Numerical experiments suggest that random illumination essentially alleviates most, if not all, numerical problems commonly associated with the standard phasing algorithms.
    • Article
    • Peer Reviewed
    Faculty (2016)
    In this paper Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) is developed for spectral estimation with single-snapshot measurement. Stability and resolution analysis with performance guarantee for Single-Snapshot ESPRIT (SS-ESPRIT) is the main focus. In the noise-free case, exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise and under the assumption that the true frequencies are separated by at least two times Rayleigh's Resolution Length, an explicit error bound for frequency reconstruction is given in terms of the dynamic range and the separation of the frequencies. The separation and sparsity constraint compares favorably with those of the leading approaches to compressed sensing in the continuum.
      • Article
      • Peer Reviewed
      Faculty (2010)
      This paper studies the problem of exact localization of sparse (point or extended) objects with noisy data. The crux of the proposed approach consists of random illumination. Several recovery methods are analyzed: the Lasso, BPDN and the One-Step Thresholding (OST). For independent random probes, it is shown that both recovery methods can localize exactly $s=\cO(m)$, up to a logarithmic factor, objects where $m$ is the number of data. Moreover, when the number of random probes is large the Lasso with random illumination has a performance guarantee for superresolution, beating the Rayleigh resolution limit. Numerical evidence confirms the predictions and indicates that the performance of the Lasso is superior to that of the OST for the proposed set-up with random illumination.
        • Article
        • Peer Reviewed
        Faculty (2012)
        This paper proposes a general framework for compressed sensing of constrained joint sparsity (CJS) which includes total variation minimization (TV-min) as an example. TV- and 2-norm error bounds, independent of the ambient dimension, are derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit. As an application the results extend Cand`es, Romberg and Tao's proof of exact recovery of piecewise constant objects with noiseless incomplete Fourier data to the case of noisy data.
          • Article
          • Peer Reviewed
          Faculty (2015)
          A brief survey of the author and collaborators' work in compressive sensing applications to continuous imaging models.
            • Article
            • Peer Reviewed
            Faculty (2003)
            We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation coefficient in the propagation direction. We discuss the limits of vanishing inner scale and divergent outer scale of the turbulent medium.
              • Thesis
              • Peer Reviewed
              UC Davis Electronic Theses and Dissertations (2024)

              This dissertation comprises two research projects focused on inverse problems in graph models, each addressing a specific application. The central theme is inverse graph problems, specifically the challenge of recovering unknown network parameters that define the underlying network structure, using only nodal measurements. In both projects, we provide theoretical guarantees for parameter recovery under certain measurement conditions.

              The first project, detailed in the first chapter, presents theoretical results on the inverse problem for power grids. We demonstrate that, given full access to phasor measurements and power injection data at all nodes, it is possible to recover sparse nodal susceptance based on the simplified AC power flow model. This recovery is achievable as long as the number of measurements exceeds a specified lower bound, a result that has not been extensively explored in the context of AC power grids.

              The second project, presented in the second and third chapters, addresses the inverse problem of parameter learning in black box neural networks. Specifically, we investigate a layer-by-layer approach for learning parameters in single-hidden-layer feedforward neural networks, focusing on networks with rectified/monotone and binary activation functions. In addition to providing theoretical guarantees for parameter recovery, we also explore algorithm design and test various algorithms on small-scale networks.

                • Article
                • Peer Reviewed
                Faculty (2002)
                We prove that the passive scalar field in the Ornstein-Uhlenbeck velocity field with wave-number dependent correlation times converges, in the white-noise limit, to that of Kraichnan's model with higher spatial regularity.
                  • Article
                  • Peer Reviewed
                  Faculty (2011)
                  Highly coherent sensing matrices arise in discretization of continuum problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold as well as in using highly coherent, redundant dictionaries as sparsifying operators. Algorithms (BOMP, BLOOMP) based on techniques of band exclusion and local optimization are proposed to enhance Orthogonal Matching Pursuit (OMP) and deal with such coherent sensing matrices. BOMP and BLOOMP have provably performance guarantee of reconstructing sparse, widely separated objects {\em independent} of the redundancy and have a sparsity constraint and computational cost similar to OMP's. Numerical study demonstrates the effectiveness of BLOOMP for compressed sensing with highly coherent, redundant sensing matrices.
                    • Article
                    • Peer Reviewed
                    Faculty (2003)
                    We consider 6 types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by various transport equations depending on the specific scaling involved. We obtain 6 different radiative transfer equations as limits.
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