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Amit Singer

  1. The Spectrum of Random Inner-product Kernel Matrices.

    Thu, 02/16/2012 - 15:01 — SomNambul
    Authors: Amit Singer, Xiuyuan Cheng
    Subjects: Probability
    Abstract

    We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1,
    ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a
    real-valued function. The eigenvalue distribution of these random kernel
    matrices is studied at the "large p, large n" regime. It is shown that, when p
    and n go to infinity, p/n = \gamma which is a constant, and f is properly
    scaled so that Var(f(X_i^T X_j)) is O(p^{-1}), the spectral density converges
    weakly to a limiting density on R. The limiting density is dictated by a cubic
    equation involving its Stieltjes transform.

  2. Vector Diffusion Maps and the Connection Laplacian.

    Wed, 02/02/2011 - 16:01 — SomNambul
    Authors: Amit Singer, Hau-Tieng Wu
    Subjects: Statistics
    Abstract

    We introduce {\em vector diffusion maps} (VDM), a new mathematical framework
    for organizing and analyzing massive high dimensional data sets, images and
    shapes. VDM is a mathematical and algorithmic generalization of diffusion maps
    and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and
    Laplacian eigenmaps. While existing methods are either directly or indirectly
    related to the heat kernel for functions over the data, VDM is based on the
    heat kernel for vector fields.

  3. Angular Synchronization by Eigenvectors and Semidefinite Programming.

    Mon, 11/23/2009 - 16:01 — SomNambul
    Authors: Amit Singer
    Subjects: Spectral Theory
    Abstract

    The angular synchronization problem is to obtain an accurate estimation (up
    to a constant additive phase) for a set of unknown angles
    $\theta_1,...,\theta_n$ from $m$ noisy measurements of their offsets
    $\theta_i-\theta_j \mod 2\pi$. Of particular interest is angle recovery in the
    presence of many outlier measurements that are uniformly distributed in
    $[0,2\pi)$ and carry no information on the true offsets. We introduce an
    efficient recovery algorithm for the unknown angles from the top eigenvector of
    a specially designed Hermitian matrix.

  4. Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I - The intrinsic reconstitution algorithm.

    Mon, 10/19/2009 - 13:48 — SomNambul
    Authors: Ronny Hadani, Amit Singer
    Subjects: Representation Theory
    Abstract

    In this paper, we describe and study a mathematical framework for
    cryo-elecron microscopy. The main result, is a a proof of the admissability
    (correctness) and the numerical stability of the intrinsic reconstitution
    algorithm which was introduced by Singer and Shkolnisky in [7]. In addition, we
    explain how the various numerical observations reported in that work, follow
    from basic representation theoretic principles.

  5. Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I - The intrinsic reconstitution algorithm.

    Mon, 10/19/2009 - 13:48 — SomNambul
    Authors: Ronny Hadani, Amit Singer
    Subjects: Representation Theory
    Abstract

    In this paper, we describe and study a mathematical framework for
    cryo-elecron microscopy. The main result, is a a proof of the admissability
    (correctness) and the numerical stability of the intrinsic reconstitution
    algorithm which was introduced by Singer and Shkolnisky in [7]. In addition, we
    explain how the various numerical observations reported in that work, follow
    from basic representation theoretic principles.

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