Integrable Generalized Principal Chiral Models

We study 2D non-linear sigma models on a group manifold with a special form of the metric. We address the question of integrability for this special class of sigma models. We derive two algebraic conditions for the metric on the group manifold. Each solution of these conditions defines an integrable model. Although the algebraic system is overdetermined in general, we give two examples of solutions. We find the Lax field for these models and calculate their Poisson brackets. We also obtain the renormalization group (RG) equations, to first order, for the generic model. We solve the RG equations for the examples we have and show that they are integrable along the RG flow.Comment: 14 pages, harvmac (l

Flat Connections for Characters in Irrational Conformal Field Theory

Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find that the affine-Virasoro characters of ICFT satisfy heat-like differential equations with flat connections. As a first example, we solve the system for the general $g/h$ coset construction, obtaining an integral representation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple $g$, noting a simplification for the subspace of theories which possess a non-trivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally-extended loop group.Comment: harvmac (answer b to question) 40 pages. LBL-35718, UCB-PTH-94/1

The finite harmonic oscillator and its associated sequences

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained. Keywords: Weil representation, commutative subgroups, eigenfunctions, random behavior, deterministic constructionComment: Published in the Proceedings of the National Academy of Sciences of the United States of America (Communicated by Joseph Bernstein, Tel Aviv University, Tel Aviv, Israel

The Beltrami Flow over Manifolds

In many medical computer vision tasks, the relevant data is attached to a specific tissue such as the cortex or the colon. This situation calls for regularization techniques which are defined over non flat surfaces. We introduce in this paper the Beltrami flow over manifolds. This new regularization technique overcomes the over-smoothing of the L_2 flow and the staircasing effects of the L_1 flow, that were recently suggested via the harmonic map methods. The key of our approach is first to clarify the link between the intrinsic Polyakov action and the implicit Harmonic energy functional and then use the geometrical understanding of the Beltrami Flow to generalize it to images on explicitly and implicitly defined non flat surfaces. It is shown that once again the Beltrami flow interpolates between the L_2 and L_1 flows on non-flat surfaces. The implementation scheme of this flow is presented and various experimental results obtained on a set of various real images illustrate the performances of the approach as well as the differences with the harmonic map flows. This extension of the Beltrami flow to the case of non flat surfaces opens new perspectives in the regularization of noisy data defined on manifolds

Learning Big (Image) Data via Coresets for Dictionaries

Signal and image processing have seen an explosion of interest in the last few years in a new form of signal/image characterization via the concept of sparsity with respect to a dictionary. An active field of research is dictionary learning: the representation of a given large set of vectors (e.g. signals or images) as linear combinations of only few vectors (patterns). To further reduce the size of the representation, the combinations are usually required to be sparse, i.e., each signal is a linear combination of only a small number of patterns. This paper suggests a new computational approach to the problem of dictionary learning, known in computational geometry as coresets. A coreset for dictionary learning is a small smart non-uniform sample from the input signals such that the quality of any given dictionary with respect to the input can be approximated via the coreset. In particular, the optimal dictionary for the input can be approximated by learning the coreset. Since the coreset is small, the learning is faster. Moreover, using merge-and-reduce, the coreset can be constructed for streaming signals that do not fit in memory and can also be computed in parallel. We apply our coresets for dictionary learning of images using the K-SVD algorithm and bound their size and approximation error analytically. Our simulations demonstrate gain factors of up to 60 in computational time with the same, and even better, performance. We also demonstrate our ability to perform computations on larger patches and high-definition images, where the traditional approach breaks down

Principal models on a solvable group with nonconstant metric

Field equations for generalized principle models with nonconstant metric are derived and ansatz for their Lax pairs is given. Equations that define the Lax pairs are solved for the simplest solvable group. The solution is dependent on one free variable that can serve as the spectral parameter. Painleve analysis of the resulting model is performed and its particular solutions are foundComment: 8 pages, Latex2e, no figure