Rachel A. Ward

Selected publications

• Sparse Legendre expansions via l1-minimization
  (with Holger Rauhut). Journal of Approximation Theory, to appear.
• Computing the confidence levels for a root-mean-square test of goodness-of-fit
  (with Will Perkins and Mark Tygert). Applied Mathematics and Computation, Volume 217, Issue 22 (2011).
• New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property
  (with Felix Krahmer). SIAM Journal on Mathematical Analysis, Volume 43 (2011).
• Quiet sigma delta quantization
  Nonlinearity, Volume 23, Number 9 (2010).
• Compressed sensing with cross validation.
  IEEE Transactions on Information Theory, Volume 55, Issue 12 (2009).

Preprints

• Stable image reconstruction using total variation minimization
  (with Deanna Needell)
• Two-subspace projection method for coherent overdetermined systems
  (with Deanna Needell)
• Chi-square and classical exact tests often wildly misreport significance; the remedy lies in computers.
  (with Will Perkins and Mark Tygert)
• Weighted eigenfunction estimates with applications to compressed sensing.
  (with Nicolas Burq, Semyon Dyatlov, and Maciej Zworski)
• Stability for second-order chaotic sigma delta quantization.
  (with Lauren Bandklayder)
• Computing the confidence levels for a root-mean-square test of goodness-of-fit, II.
  (with Will Perkins and Mark Tygert)
• A symbol-based bar code decoding algorithm.
  (with Mark Iwen and Fadil Santosa)

Talks

• Chi-square and classical exact tests often wildly misreport significance; the remedy lies in computers

  ACM Colloquium, Caltech, 10/2011.pdf >
  (Joint work with Will Perkins and Mark Tygert) Pearson's chi-square statistic for discrete goodness-of-fit testing overweights bins with small expected frequencies in order to fix the asymptotic distribution to be chi-square. This often leads to inconsistent and inaccurate reporting of significance levels in practice. However, with the now widespread availability of computers, having a fixed asymptotic distribution is no longer necessary for computation; this affords the freedom to design more meaningful goodness-of-fit tests. In the paper of the same name, we show via numerous examples that a simple unweighted root-mean-square statistic, whose significance levels can be computed using Monte-Carlo simulation, remedies many of the problems with classic goodness-of-fit tests. You can read more here. In the papers Computing the confidence levels for a root-mean-square test of goodness-of-fit, Parts I and II, we give fast black-box algorithms for computing the (distribution-dependent) asymptotic distribution of this root-mean-square statistic.

• Near-equivalence of the Restricted Isometry Property and Johnson-Lindenstrauss Lemma

  Workshop in Probabilistic Reasoning in Quantitative Geometry, MSRI, 9/2011.pdf >
  (Joint work with Felix Krahmer) This talk is on the paper, New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property. In short, we provide the converse to the statement "JL implies RIP" established in A Simple Proof of the Restricted Isometry Property for Random Matrices by Baraniuk, Davenport, DeVore, and Wakin: we show that randomizing the column signs of any matrix with the Restricted Isometry Property produces a Johnson-Lindenstrauss embedding. In other words, by randomizing the column signs of a matrix that acts as a near-isometry on k-sparse vectors, one produces a matrix which with high probability acts as a near-isometry on any fixed set of 2^k vectors. This result gives improved bounds on the minimal dimension in which one can embed points near-isometrically using structured random maps which have the advantage of fast matrix-vector multiplication algorithms.

• Sparse recovery for spherical harmonic expansions

  Workshop Sparsity and Cosmology, Nice, 5/2011.pdf >
  (Joint work with Holger Rauhut) A foundational result in compressed sensing is that sparse expansions in a given basis can be efficiently recovered from a small number of nonadaptive linear measurements if the measurements are incoherent with respect to the sparsity basis. For example, signals that are sparse in frequency can be recovered from a few randomly-selected pointwise measurements; signals that are sparse in the standard basis can be recovered from a few randomly-selected DFT measurements. In the papers Sparse Legendre expansions via l1 minimization and Sparse recovery for spherical harmonic expansions, we show that for optimal sparse recovery results, the strict notion of incoherence between sparsity basis and sampling distribution can be relaxed using certain weighted L-infinity estimates. For example, sparse spherical harmonic expansions can be recovered from near-optimally few samples by drawing sampling points from a certain fractional tangent measure, even though the spherical harmonic basis is not uniformly bounded. This last result, not included in the above talk, is joint with Nicolas Burq, Semyon Dyatlov, and Maciej Zworski.

Collaborators

• Boris Alexeev @Princeton
• Lauren Bandklayder @University of Münster
• Nicolas Burq @Universite Paris-Sud
• Semyon Dyatlov @Berkeley
• Massimo Fornasier @TU München
• Mark Iwen @Duke
• Felix Krahmer @Göttingen
• Deanna Needell @Claremont McKenna
• Will Perkins @Georgia Tech
• Holger Rauhut @Hausdorff Center, Bonn
• Rayan Saab @Duke
• Fadil Santosa @University of Minnesota
• Mark Tygert @Courant Institute
• Maciej Zworski @Berkeley

Teaching

• May 2011: Sparsity and Computation, Program for Women and Mathematics, Institute for Advanced Study, Princeton.
• Spring 2010/ Fall 2010: Calculus III
• Summer 2009: Discrete dynamical systems with applications to A/D conversion, Summer Workshop In Mathematics (SWIM) for high school students, Princeton University.
• Fall 2008: Integrated Math, Engineering, and Physics (similar to Calculus III), Princeton University.

Calendar