Some closure features of locally testable affine-invariant properties

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 31-32).We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less well-studied property, which creates some interesting affine-invariant properties over large domains, from properties over smaller domains. Previously such results were known for "single-orbit characterized" affine-invariant properties, which are known to be a subclass of locally testable ones, and are potentially a strict subclass. The fact that the intersection of locally-testable affine-invariant properties are locally testable could have been derived from previously known general results on closure of property testing under set-theoretic operations, but was not explicitly observed before. The closure under sum and lifts is implied by an affirmative answer to a central question attempting to characterize locally testable affine-invariant properties, but the status of that question remains wide open. Affine-invariant properties are clean abstractions of commonly studied, and extensively used, algebraic properties such linearity and low-degree. Thus far it is not known what makes affine-invariant properties locally testable - no characterizations are known, and till this work it was not clear if they satisfied any closure properties. This work shows that the class of locally testable affine-invariant properties are closed under some very natural operations. Our techniques use ones previously developed for the study of "single-orbit characterized" properties, but manage to apply them to the potentially more general class of all locally testable ones via a simple connection that may be of broad interest in the study of affine-invariant properties.by Alan Xinyu Guo.S.M

Lower bounds for constant query affine-invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well-suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally correctable code (LCC), where $\mathbb{K}$ is a finite field and $\Sigma$ is a finite alphabet, then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-1}))$. Also, we show that if $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally testable code (LTC), then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-2}))$. The dependence on $n$ in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS `13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto a small error in the Gowers norm

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

Chern-Simons Gravity: From 2+1 to 2n+1 Dimensions

These lectures provide an elementary introduction to Chern Simons Gravity and Supergravity in $d=2n+1$ dimensions.Comment: 17 pages, two columns, latex, no figures, Lectures presented at the XX Encontro de Fisica de Particulas e Campos, Sao Lourenco, Brazil, October 1999, and at the Fifth La Hechicera School, Merida, Venezuela, November 199

Gauge Theories of Gravitation

During the last five decades, gravity, as one of the fundamental forces of nature, has been formulated as a gauge theory of the Weyl-Cartan-Yang-Mills type. The present text offers commentaries on the articles from the most prominent proponents of the theory. In the early 1960s, the gauge idea was successfully applied to the Poincar\'e group of spacetime symmetries and to the related conserved energy-momentum and angular momentum currents. The resulting theory, the Poincar\'e gauge theory, encompasses Einstein's general relativity as well as the teleparallel theory of gravity as subcases. The spacetime structure is enriched by Cartan's torsion, and the new theory can accommodate fermionic matter and its spin in a perfectly natural way. This guided tour starts from special relativity and leads, in its first part, to general relativity and its gauge type extensions \`a la Weyl and Cartan. Subsequent stopping points are the theories of Yang-Mills and Utiyama and, as a particular vantage point, the theory of Sciama and Kibble. Later, the Poincar\'e gauge theory and its generalizations are explored and special topics, such as its Hamiltonian formulation and exact solutions, are studied. This guide to the literature on classical gauge theories of gravity is intended to be a stimulating introduction to the subject.Comment: 169 pages, pdf file, v3: extended to a guide to the literature on classical gauge theories of gravit

Noncommutative Geometries and Gravity

We briefly review ideas about ``noncommutativity of space-time'' and approaches toward a corresponding theory of gravity.Comment: 18 pages, to appear in the Proceedings of the Third Mexican Meeting on Mathematical and Experimental Physics, Symposium on Gravitation and Cosmology, Mexico City, 10-14 September 200

Deriving the mass of particles from Extended Theories of Gravity in LHC era

We derive a geometrical approach to produce the mass of particles that could be suitably tested at LHC. Starting from a 5D unification scheme, we show that all the known interactions could be suitably deduced as an induced symmetry breaking of the non-unitary GL(4)-group of diffeomorphisms. The deformations inducing such a breaking act as vector bosons that, depending on the gravitational mass states, can assume the role of interaction bosons like gluons, electroweak bosons or photon. The further gravitational degrees of freedom, emerging from the reduction mechanism in 4D, eliminate the hierarchy problem since generate a cut-off comparable with electroweak one at TeV scales. In this "economic" scheme, gravity should induce the other interactions in a non-perturbative way.Comment: 30 pages, 1 figur