A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Testability of linear-invariant properties

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 75-80).Property Testing is the study of super-efficient algorithms that solve "approximate decision problems" with high probability. More precisely, given a property P, a testing algorithm for P is a randomized algorithm that makes a small number of queries into its input and distinguishes between whether the input satisfies P or whether the input is "far" from satisfying P, where "farness" of an object from P is measured by the minimum fraction of places in its representation that needs to be modified in order for it to satisfy P. Property testing and ideas arising from it have had significant impact on complexity theory, pseudorandomness, coding theory, computational learning theory, and extremal combinatorics. In the history of the area, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, homogeneousness, Reed-Muller codes, and Fourier sparsity. In this thesis, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory. We also show the first nontrivial lowerbound for the query complexity of a natural testable linear-invariant property.by Arnab Bhattacharyya.Ph.D

Multi-instantons and supersymmetric SU(N) gauge theories

In this thesis the proposed exact results for low energy effective N = 2 supersymmetric SU(N) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets in four dimensions are considered. The proposed exact results are based upon the work of Seiberg and Witten for low energy effective four dimensional M = 2 supersymmetric SU[2) Yang-Mills gauge theory coupled to Nf fundamental matter multiplets. The testing and matching of the proposed exact results via supersymmetric instanton calculus are the motivation for two studies. Firstly, we study the ADHM construction of instantons for gauge groups U(N) and SU(2) and for topological charge two and three. The ADHM constraints which implicitly specify instanton gauge field configurations are solved for the explicit exact general form of instantons with topological charge two and gauge group U[N). This is the first explicit and general multi-instanton configuration for the unitary gauge groups. The U[N) ADHM two-instanton configuration may be used in further tests and matching of the proposed exact results in low energy effective M =2 supersymmetric SU(iV) Yang-Mills gauge theories by comparison with direct instanton calculations. Secondly, a one-instanton level test is performed for the reparameterization scheme proposed by Argyres and Pelland matching the conjectured exact low energy results and instanton predictions for the instanton contributions to the prepotential of low energy effective M = 2 supersymmetric SU [N) Yang-Mills gauge theory with Nf = 2N mass-less fundamental matter multiplets. The constants within the reparameterization scheme which ensure agreement between the exact results and the instanton predictions for general N > 1 are derived for the entire quantum moduli space. This constitutes a non-trivial test of the proposed reparameterization scheme, which eliminates the discrepancies arising when the two sets of results are compared

Topics in perturbation theory

In providing a means of progressively improving an initial estimate, perturbation series have become a ubiquitous tool in modern physics. However, and mainly because this stepwise process of improvement rapidly becomes increasingly involved, surprisingly little is known about the formal properties of the series obtained. This thesis therefore investigates some aspects of these properties and how they effect the application of these techniques, with an emphasis on quantum field theory and the phenomenology of e+e(^-) colliders. One of the better understood examples of a perturbative series is the WKB one which is widely used to approximate the energy levels of quantum mechanical systems. Recently much interest has centred on a modification of this, the SWKB series. Apart from (possibly) offering an improvement on the original, this is intrinsically interesting in being related to the supersymmetry of field theory. Furthermore, as Chapter 1 explains, there is a close connection between the cases where the initial estimate requires no correction and an important set of quantum mechanical problems (the "shape invariant" ones) which can be solved elegantly and completely. The situation in field theory is more complicated, not least because the series for any particular problem is no longer unique. While this presents few theoretical difficulties, it has serious consequences when attempts are made to compare predictions with experiment. This obstacle is particularly severe in Quantum Chromodynamics and its fundamental constant (A(_QCD)) is therefore only roughly known at present. It will be argued that current responses to this are all imperfect, but that tests of the theory can be envisaged that circumvent the problem. This leads into questions concerning the origin of the divergences in the perturbation series - for although it may initially provide usefully improved estimates, the series probably breaks down eventually. Existing arguments about this topic are critically reviewed - and in one case substantially simplified - before an alternative one is proposed in some detail. By concentrating on a particularly restricted situation, the Common Effective Charge Approach simplifies matters to the extent that issues such as non-analyticity of functions and the potential accuracy of perturbative techniques in realistic applications can be conveniently investigated

Optimization and the convergence of perturbation series

This thesis is concerned with the possible sums of perturbation series in mass- less, renormalizable field theories. It shows that, given a free choice of scheme, the limit of the sequence of approximants is arbitrary. Restricting the choice to finite schemes, in particular "zero schemes", yields a perturbatively unique limit to the sequence of approximants. An operational method for calculating perturbative expansions in the class of zero schemes is discussed. A comparison of various optimization schemes is given for a few phenomenological examples in QCD and QED

Some investigations into the structure of local gauge-invariant field theories

Imperial Users onl

Single mode excitation in the shallow water acoustic channel using feedback control

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution June 1996The shallow water acoustic channel supports far-field propagation in a discrete set of modes. Ocean experiments have confirmed the modal nature of acoustic propagation, but no experiment has successfully excited only one of the suite of mid-frequency propagating modes propagating in a coastal environment. The ability to excite a single mode would be a powerful tool for investigating shallow water ocean processes. A feedback control algorithm incorporating elements of adaptive estimation, underwater acoustics, array processing and control theory to generate a high-fidelity single mode is presented. This approach also yields a cohesive framework for evaluating the feasibility of generating a single mode with given array geometries, noise characteristics and source power limitations. Simulations and laboratory waveguide experiments indicate the proposed algorithm holds promise for ocean experiments.Josko Catipovic funded my research for summer of 1992 on the Office of Naval Research Grant Number N00014-92-J-1661 and from June 1993 through August 1995 on Defense Advanced Research Projects Agency Grant Number MDA972-92-J- 1041. The Office of Naval Research Grant N00014-95-1-0362 to MIT supported the computer facilities used to do much of this work

Braneworld black holes

The braneworld paradigm provides an interesting framework within which to explore the possibility that our Universe lives in a fundamentally higher dimensional space- time. In this thesis we investigate black holes in the Randall-Sundrum braneworld scenario. We begin with an overview of extra-dimensional physics, from the original proposal of Kaluza and Klein up to the modern braneworld picture of extra dimensions. A detailed description of braneworld gravity is given, with particular emphasis on its compatibility with experimental tests of gravity. We then move on to a discussion of static, spherically symmetric braneworld black hole solutions. Assuming an equation of state for the "Weyl term", which encodes the effects of the extra dimension, we are able to classify the general behaviour of these solutions. We then use the strong field limit approach to investigate the gravitational lensing properties of some candidate braneworld black hole solutions. It is found that braneworld black holes could have significantly different observational signatures to the Schwarzschild black hole of standard general relativity. Rotating braneworld black hole solutions are also discussed, and we attempt to generate rotating solutions from known static solutions using the Newman-Janis complexification "trick"